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## Functions and Analysis

### Basic Functions - Polynomial Functions

#### Q.02

'Examine the rate of change of the following functions and find the extremum. (1) $y=\x0crac{1}{3} x^{3}-9 x$ (2) $y=-x^{3}+x^{2}-x+1$'

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'Find the equation of the tangent line drawn from the point (0,4) to the curve y=x^3+2.'

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'When a = 0, b can be any real number. When a ≠ 0, b < 0 and b < a³ / 27 or b > 0 and b > a³ / 27.'

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'What is the condition for the graph of a function y = f(x) to intersect the positive part of the x-axis at two different points?'

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'For the curve C: y=x^{3}+3 x^{2}+x, when there are 3 tangents passing through point A(1, a), find the range of constant a.'

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'(2) 2-c ≤ 2 ≤ 2+c (1). Therefore\n \\[ \egin{aligned} P(2-c ≤ X ≤ 2+c) & =\\int_{2-c}^{2+c} f(x) d x \\ & =\\int_{2-c}^{2}(x-1) d x-\\int_{2}^{2+c}(x-3) d x \\ & =\\left[\\frac{(x-1)^{2}}{2}\\right]_{2-c}^{2}-\\left[\\frac{(x-3)^{2}}{2}\\right]_{2}^{2+c} \\ & =-(c-1)^{2}+1 \\end{aligned} \\] \n Therefore, when P(2-c ≤ X ≤ 2+c)=0.5\n \\[ -(c-1)^{2}+1=0.5 \\text{ i.e. } \\quad(c-1)^{2}=\\frac{1}{2} \\] \n Solving this, c-1= \\pm \\frac{1}{\\sqrt{2}} which leads to \\quad c=\\frac{2 \\pm \\sqrt{2}}{2} \n When c=\\frac{2+\\sqrt{2}}{2}, (1) becomes, 1-\\frac{\\sqrt{2}}{2} ≤ X ≤ 3+\\frac{\\sqrt{2}}{2}, contradicting 1 ≤ X ≤ 3. When c=\\frac{2-\\sqrt{2}}{2}, (1) becomes, 1+\\frac{\\sqrt{2}}{2} ≤ X ≤ 3-\\frac{\\sqrt{2}}{2}, satisfying 1 ≤ X ≤ 3. Therefore \\quad c=\\frac{2-\\sqrt{2}}{2}'

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'The graph of y = f(t) looks as shown on the right. Therefore, f(t) takes the minimum value of (2-√2)/6 at t = √2/4.'

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"Example 74\n(1) From the condition f(x) = ∫(2x^2 - 3x) dx = (2/3)x^3 - (3/2)x^2 + C\nGiven f(0) = 2, C = 2\nTherefore, f(x) = (2/3)x^3 - (3/2)x^2 + 2\n(2) The slope of the tangent line at point (x, f(x)) on the curve y = f(x) is f'(x)\nSo, f'(x) = x^2 - 1\nHence, f(x) = ∫(x^2 - 1) dx = (1/3)x^3 - x + C\n(C is the constant of integration)\nThe curve y = f(x) passes through the point (1,0) so f(1) = 0\nThus, (1/3) - 1 + C = 0\nTherefore, C = 2/3\nTherefore, f(x) = (1/3)x^3 - x + (2/3)"

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"Solve the equation f(x)=0 to obtain the real solutions x=-1 and x>2, thus there is 1 positive solution and 1 negative solution. Tilt 72, see this book p.293. Rearrange the equation f(x)=-2x^3+6x to get -2x^3+6x=a, f'(x)=-6x^2+6=-6(x+1)(x-1). Solving f'(x)=0 gives x=±1, and the increasing and decreasing table of f(x) is shown in the table below. Therefore, the graph of y=f(x) is as shown on the right. The number of real solutions of the equation f(x)=a is determined by the number of intersection points between the graph of y=f(x) and the line y=a, resulting in 1 solution when a<-4 or a>4, 2 solutions when a=-4 or a=4, and 0 solutions when -4<a<4. The point (-1,0) is the tangent point of the graph and the x-axis. The value of a when the line y=a passes through the maximum and minimum points of the function is the boundary for the number of real solutions. Example"

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'Practice 175 → this workbook p .324\n(1) When the curve y=f(x) and the line y=mx+n are tangent at two points x=a, b(a<b), the following identity holds true.'

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'Practice, where a, b are constants, 0<a<1. Find the values of a and b such that the function f(x)=x^{3}+3 a x^{2}+b (-2 ≤ x ≤ 1) has a maximum value of 153 and a minimum value of -5.'

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'Please plot the graphs of the following functions.'

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'Solve the equation $x+2 \\underset{2^{\x0crac{1}{2}} x^{-\x0crac{1}{2}}}{\\hookrightarrow 1} + x^{-1}=5$'

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'Let 72a be a real number. Two lines with slopes m are tangent to the curve y=x^{3}-3 a x^{2} at points A and B, respectively.'

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'−8 − 6√2 ≤ x²y + xy² − x² − 2xy − y² + x + y ≤ 3'

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'Create a table of increasing and decreasing intervals for f(x) = x^{4} - 6 x^{2} - 8 x - 3, and determine the number of real solutions.'

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'The x-coordinates of the intersection points of the curve y=f(x) and the parabola y=h(x) are obtained by solving the equation x^4 - 2x^2 + 4x = -x^2 + 4x, which simplifies to x^4 - x^2 = 0, thus giving x = 0, ±1.'

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'Find the extreme values of the function f(x) = x^3 - 3x^2 + 2x.'

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'Find the maximum and minimum values of the function g(x) = x^4 - 4x^3 + 4x^2.'

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'Given real numbers α, β, and γ satisfying α+β+γ=3, let p=αβ+βγ+γα, q=αβγ. Prove: (1) When p=q+2, at least one of α, β, and γ is 1. (2) When p=3, prove that α, β, and γ are all 1.'

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'Find the following definite integrals. (1) $\\int_{0}^{2}\\left|x^{2}-4 x+3\\right| d x$(2) $\\int_{0}^{\\frac{\\sqrt{5}-1}{2}}\\left(x^{2}+x-1\\right) d x$'

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'Exercise 74 \ \\triangle OPQ = \\frac{1}{2}|\\cos \\theta \\cdot 3 \\sin 2 \\theta - \\sin \\theta \\cdot 1|\ \\( \egin{aligned} &=\\frac{1}{2}|\\cos \\theta \\cdot 6 \\sin \\theta \\cos \\theta - \\sin \\theta| &=\\frac{1}{2}\\left|6 \\sin \\theta\\left(1-\\sin ^{2} \\theta\\right)-\\sin \\theta\\right| &=\\frac{1}{2}\\left|-6 \\sin ^{3} \\theta + 5 \\sin \\theta\\right| \\end{aligned} \\) Let \ \\sin \\theta = t \ Then, for \ 0 \\leq \\theta < 2 \\pi \ we get \ \\left|-3 t^{3} + \\frac{5}{2} t\\right| \ Let \\( f(t) = -3 t^{3} + \\frac{5}{2} t \\) So that, \\( f^{\\prime}(t) = -9 t^{2} + \\frac{5}{2} = -9\\left(t + \\sqrt{\\frac{5}{18}}\\right)\\left(t - \\sqrt{\\frac{5}{18}}\\right) = -9\\left(t + \\frac{\\sqrt{10}}{6}\\right)\\left(t - \\frac{\\sqrt{10}}{6}\\right) \\) \\[ f^{\\prime}(t) = 0 \\] Solving this equation gives \ t == \\pm \\frac{\\sqrt{10}}{6} \ The variation table of \\( f(t) \\) for \ -1 \\leqq t \\leqq 1 \ is as follows.'

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'Find the x-coordinate of the point of tangency where the slope of the tangent to the curve y = x^{3} - 3x^{2} is 9.'

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'Find the number of real solutions of f(x)=x^{3}-6 x^{2}+9 x-5.'

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'Practice: Two distinct points P, Q on the curve C: y=x^{3}-mx are on the curve with respect to the origin O. If the tangent line at point Q on C is parallel to the line OP, then: (1) If the x-coordinate of P is a, express the x-coordinate of Q using a. (2) Find the range of values for m for angle POQ to be a right angle. [Shimane University] => p. 300 Exercises 69'

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'Let a be a constant, where a>1. For the function y=2x^3-9x^2+12x with 1 ≤ x ≤ a, find the following values: (1) minimum value (2) maximum value'

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'When 0<a<2, from the graph on the right, the maximum value of f(a)=-a^{3}+3a^{2} is achieved at x=a. When 2 ≤ a, from the graph on the right, the maximum value of f(2)=4 is achieved at x=2. When 0<a<2, the maximum value is achieved at x=a for -a^{3}+3a^{2}, and when 2 ≤ a, the maximum value of 4 is achieved at x=2.'

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'(3) \\( (x+2 y-4)\\left(x^{2}+y^{2}-2 x-8\\right)<0 \\)'

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'Find the value of the constant $a$ when the curves $y=x^{3}-2 x+1$ and $y=x^{2}+2 a x+1$ are tangent. Also, determine the equation of the common tangent line at the point of tangency.'

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'Find the value of the constant a when the difference between the maximum and minimum values of the function f(x)=x^{3}-3 x^{2}+3 a x-2 is 32.'

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'Find the range of constant $k$ for which the equation $x^{3}+5 x^{2}+3 x+k=0$ has one positive root and two distinct negative roots.'

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'Practice problem 8 Find the area between the graphs of two cubic functions'

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'In the basic example 190, from the selection of maximum and minimum values, a, b are constants, and a>0. For the function f(x) = a x^{3} - 9 a x^{2} + b, (1) determine the maximum and minimum values in the interval -1 ≤ x ≤ 3 in terms of a, b. (2) Determine the values of a, b such that the maximum value in (1) is 10, and the minimum value is -44.'

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'For the TR curve y=x^{2}-3 x+2, find the equations of the following tangents:\n(1) Tangent at point (3,2) on the curve\n(2) Tangent with a slope of -1'

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'Let l be the line 2x+y+2=0 and P be a point on the parabola y=x^2. Find the coordinates of P when the distance between P and l is minimized. Also, calculate the distance of Pl at that moment.'

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'Maximum and minimum of a cubic function containing 189 characters'

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'Find the maximum and minimum values of the given functions.'

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'Find the slope of the line that forms an angle of π/4 with the line x - √3 y = 0.'

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'Sketch the region represented by the following inequalities: (1) \\left\\{\egin{\overlineray}{l}x-3 y-9<0 \\\\ 2 x+3 y-6>0\\end{\overlineray}\\right. (2) \\left\\{\egin{\overlineray}{l}x^{2}+y^{2} \\leqq 9 \\\\ x-y<2\\end{\overlineray}\\right. (3) $1<x^{2}+y^{2} \\leqq 4$'

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'37 Function Growth and Decrease · Application of Graphs Standard 183 Number of Real Solutions of Cubic Equations (2) f(x) = constant'

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'Provide an example of a curve passing through the point (0,1).'

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'Determine the range of values for the constant a so that the function f(x)=x^{3}+ax^{2}+(3a-6)x+5 has maximum and minimum values.'

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'Given that 189 is a constant and a>0. Find the maximum value of the function f(x)=-x^{3}+3ax(0 ≤ x ≤ 1).'

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'Standard 64: Determination of Coefficients of Higher-order Equations (1) - Conditions for Real Solutions'

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'In mathematical proof by induction, the central part is part [2]. It is crucial to clearly understand the assumption when n=k and the conclusion when n=k+1 (what you want to prove), and the key is in how to logically deduce the conclusion from the assumption.'

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'Increase and decrease of functions and maximum/minimum basics 177 3rd-degree function increase and decrease'

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'Find the following definite integral. (1) $\\int_{-1}^{2}\\left(2 x^{2}-x+3\\right) d x$'

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'If x+y+z=1/x+1/y+1/z=1, prove that at least one of x, y, z is 1.'

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'Let a be a constant, where a>0. For the function f(x)=x^{3}-3 a^{2} x (0 ≤ x ≤ 1):\n(1) Find the minimum value.\n(2) Find the maximum value.'

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'The cubic function f(x)=a x^{3}+b x+3 has a local minimum of 1 at x=-1. Determine the values of constants a and b. Also, find the maximum value.'

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'37. Increase and decrease of functions, application of graphs, basic 182, (1) Basic of the number of real solutions of cubic equations'

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'Prove that the equation (a+b)(b+c)(c+a)+abc=0 holds when (2) a+b+c=0.'

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'Determining the coefficients of a cubic function from the conditions of extrema'

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'Find the sum of the areas of the two shapes enclosed by the curves y=x^3-4x and y=3x^2.'

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'Let a be a constant, where a>0. Find the maximum value of the function f(x)=-x^{3}+3ax(0 ≤ x ≤ 1).'

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"Question 8: The contrast between the dimly lit board house during the day and the rose shining in bright colors, as well as the darkness of the night in Tsujido contrasting with the moonlit sandy ground, depicts a striking beauty, making option E the best choice. The withered rose does not symbolize the remaining life of the sister, so it is not suitable. Concentrating on one's appearance is an action opposite to 'lack of motivation,' so it is incorrect. The reason for the father bravely stepping into the deep puddle after scolding was because he showed his love. The intention to express 'Senichi's pettiness after losing human pride' is not evident from this. The mention of passing through the cemetery path is not suitable."

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'(6) The brightness of a star is not only determined by differences in color, even cool and red stars, if they are large in size, will appear bright.'

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'If real numbers x, y satisfy 2x^{2}+3y^{2}=1, find the maximum and minimum values of x^{2}-y^{2}+xy.'

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'For the vectors \\( \\vec{a}=(3,-4,12), \\vec{b}=(-3,0,4), \\vec{c}=\\vec{a}+t \\vec{b} \\), find the value of the real number \ t \ such that the angle between \ \\vec{c} \ and \ \\vec{a} \, and between \ \\vec{c} \ and \ \\vec{b} \ is equal.'

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'Plot the graph of the following function and determine its range.'

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'A moving point P on the curve xy = 4, when a perpendicular line PQ is drawn to the y-axis, point Q moves along the positive direction of the y-axis with a speed of 2 units per second. Find the speed and acceleration when point P passes through the point (2,2).'

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'Calculate the area of the curve represented by the polar equation.'

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'Prove the continuity of the function f(x)=\\left\\{\egin{\overlineray}{ll}x^{2} & (x \\neq 0) \\\\ 1 & (x=0)\\end{\overlineray}\\right.'

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'Find the composite function f(g(x)) for f(x)=x^{2}+x+2 and g(x)=x-1.'

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'Express a and b in terms of n when g(x)=a x^(n+1)+b x^n+1 (where n is a natural number greater than or equal to 2) is divisible by (x-1)^2.'

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'A sphere with a constant rate of increase in surface area of 4πcm^2/s. Determine the following when the radius becomes 10cm:'

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'Plot the graph of the following functions for practice 101 times. (1) y=x^{2}-3|x|+2 (2) y=|2 x^{2}-4 x-6| (3) y=|x+1|(x-2)'

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'Plot the graph of the following functions: (1) y=x^{2}-4|x|+2 (2) y=|x^{2}-4|'

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'Find the maximum and minimum values of the function y=(x^2-2x)(6-x^2+2x) when -1 ≤ x ≤ 3.'

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'A positive integer represented in decimal is converted to quaternary, resulting in a 3-digit number abc; converting it to senary results in a 3-digit number pqr. Suppose a + b + c = p + q + r. Write this number in decimal.'

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'Find the maximum and minimum values of the function y=x^4-8x^2+1.'

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'When a equals 1, f(x) reaches a minimum at x=1. Therefore, f(1)=-3a+7≥0, which implies a≤7/3. The common range between 1<a and 1<a≤7/3 is 1<a≤7/3.'

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'Understand the range of a function and conquer Example 64!'

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'Plot the graphs of the following functions and determine their ranges.'

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'Explain the reason why the same relational expression is derived in Lecture [1] and [3].'

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'Translate the given text into multiple languages.'

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"Let's solve these two quadratic inequalities using graphs. Here, we are dealing with inequalities in terms of m, not x, so the graph will be on the m-axis."

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'When 59(a, b) = (9,8), (12,6), the maximum value is 72'

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'A 6m long wire is bent at right angles to create a fence at the corner of a right-angled wall. How should the wire be bent to maximize the area of the fence?'

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'Let $0 \\leq t \\leq 1$. Find the values of $t$ that maximize and minimize the definite integral $\\int_{0}^{1}\\left|x^{2}-t^{2}\\right|dx$.'

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'Intersection of the graph of a function and its tangent\n(1)\nOn the curve C: y=x^{3}-x, there is a point A with x-coordinate 1. Find the x-coordinate of the other point where the tangent at point A intersects with C.'

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'Given the function f(x) = ax^3 - 6ax^2 + b, and the maximum value in the range -1 <= x <= 2 is 3, and the minimum value is -29, find the values of the constants a and b.'

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'When the equation $f(x)=a$ has three different real solutions, the curve $y=f(x)$ and the line $y=a$ have three different points of intersection.'

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'Find the range of values for the constant m such that the function f(x)=x^3-3mx^2+6mx has extreme values.'

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'Find the extreme values of the following functions and plot their graphs. (1) y=x^{3}-3 x (2) y=x^{3}+3 x^{2}+3 x+3'

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'Find the range of values for the constant $a$ such that the function $f(x)=x^{3}+ax^{2}+(3a-6)x+5$ has critical points.'

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'Choose one that fits in (4) E from the following 0-2.'

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'Find the minimum value of 4x^2 + 1/((x + 1)(x - 1)) when x > 1.'

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'Investigate the increasing and decreasing behavior of the function f(x)=|x|(x^2-5x+3) and sketch the general shape of the graph of y=f(x).'

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'(1) Represent the region described by the system of inequalities {x^2 + y^2 - 2x + 2y - 7 ≥ 0, x ≥ y} graphically. (2) Given r > 0. Find the maximum value of r for which the conditions (x-4)^2 + (y-2)^2 ≤ r^2 hold.'

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'Consider the line lt passing through point P: y = 2tx + t^2. Find the equation of the trajectory of point P. Also, when t changes over all real values, illustrate the set of all points (x, y) through which the line lt passes.'

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'Find the average rate of change when x varies within [ ]. (a) f(x)=-3 x^{2}+2 x from -2 to b (b) f(x)=x^{3}-x from a to a+h'

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'Find the number of tangents drawn from point (0, k) to the curve C: y=-x^3+3x^2.'

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'Find the general term of the sequence defined by the following conditions: \ a_{1}=3, a_{n+1}=2 a_{n}-n \'

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'Proof of equations and inequalities Basic principles 1. Proof of equation A=B 1. Transform one of A or B to derive the other. It is a principle to transform the more complex expression. 2. Transform A, B respectively to derive the same expression. 3. Transform A-B to show that it becomes 0. A=B ⇔ A-B=0'

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'Let a, b be real numbers. The 3rd degree function f(x)=x^{3}+a x^{2}+b x has a maximum at x=α and a minimum at x=β. Here, α<β.'

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'Prove that for real numbers x and y, if x^{2}+y^{2}<1, then x^{2}+y^{2}<2 x+3.'

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'When the graph of the cubic function y=ax^3+bx^2+cx+d looks like the one on the right, determine the signs of a, b, c, and d.'

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'Find the equation of the tangent line to the curve y=x^{3} drawn from point (1,0).'

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'When a < 0, g(a) = 2a^3 - 3a^2 + 3\nWhen 0 ≤ a < 1, g(a) = 3\nWhen 1 ≤ a < (6+√6)/6, g(a) = 2a^3 - 9a^2 + 12a - 2\nWhen (6+√6)/6 ≤ a, g(a) = 2a^3 - 3a^2 + 3'

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Join our Discord#### Q.16

'Therefore, the graph of function (1) is as shown on the right, with 3 points of intersection with the x-axis. Therefore, the equation x^{3}-3 x^{2}+1=0 has 3 real solutions.'

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'Calculate the area enclosed by the curves $y=x^{3}-4 x$ and $y=3 x^{2}$.'

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Join our Discord#### Q.19

'Find the area enclosed by the following curves, lines, and x-axis.'

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Join our Discord#### Q.20

'Find the maximum and minimum values of the given functions.'

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Join our Discord#### Q.22

'Find the minimum value of the function P=x^{2}+3y^{2}+4x-6y+2 for x and y.'

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'Answer the following questions about the graph of the function y=|x^2-2mx|-m. Here, m is a real number.'

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Join our Discord#### Q.24

'Find the maximum and minimum values of the following functions.'

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Join our Discord#### Q.25

'The parabolas y = x^2 + 1 and y = -x^2 - 1 do not have any points in common.\nHowever, investigate what equation of a line will be obtained using this method.'

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'When defining the function f(x) (0 ≤ x < 1), plot the graph of the following functions. (1) y=f(x) (2) y=f(f(x))'

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'(2) \ \\left\\{\egin{\overlineray}{l}x^{2}-y^{2}+x+y=0 \\\\ x^{2}-3 x+2 y^{2}+3 y=9\\end{\overlineray}\\right. \'

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Join our Discord#### Q.29

'For the function f(x)=x^2-2x-3, answer the following questions.'

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Join our Discord#### Q.30

'Find the maximum and minimum values of the function f(x) = |x^2 - 1| - x for -1 ≤ x ≤ 2.'

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Join our Discord#### Q.31

'Find the maximum and minimum values of the function y = (x ^ 2 - 2x - 1) ^ 2 - 6(x ^ 2 - 2x - 1) + 5 for -1 ≤ x ≤ 2.'

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Join our Discord#### Q.32

'Determine whether the function f(x) has a maximum value for the given range of a.'

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'In the function y=f(x), what is the notation for the function when the domain is a ≤ x ≤ b?'

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'Please graph the following functions. (1) y=x^{2}-4|x|+2 (2) y=\\left|x^{2}-4\\right|'

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'Plot the graph of the function y = |x^{2} - x - 2| - 2x.'

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Join our Discord#### Q.36

'Plot the graphs of the following functions and find their ranges.'

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Join our Discord#### Q.41

'Consider the necessary and sufficient conditions.'

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Join our Discord#### Q.42

'Find the maximum and minimum values of the function y=x^{4}-8x^{2}+1.'

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'When \ 1 < a \, the function \\( f(x) \\) reaches its minimum at \ x = 1 \. Therefore, \\( \\quad f(1) = -3a + 7 \\geqq 0 \\), which implies that \ a \\leqq \\frac{7}{3} \'

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'Find the maximum and minimum values of the function y = (x² - 6x)² + 12(x² - 6x) + 30 when 1 ≤ x ≤ 5.'

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'Please explain the domain and range of the function y = f(x).'

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Join our Discord#### Q.46

'Select two functions from the following (1) to (4) that have maximum values at x=2, and find the maximum and minimum values of those functions.'

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Join our Discord#### Q.47

'Convert the polar equation to rectangular coordinates:\nFrom the polar equation $r=\\frac{3}{1+2 \\cos \\theta}$ we have $r+2r \\cos \\theta=3$\nSince $r \\cos \\theta=x$, we get $r+2x=3$\nTherefore, $r=3-2x$, which implies $r^{2}=(3-2x)^{2}$\nAs $r^{2}=x^{2}+y^{2}$, we have $x^{2}+y^{2}=(3-2x)^{2}$\nSolving this equation gives $3x^{2}-12x-y^{2}+9=0$\nHence, $3(x-2)^{2}-y^{2}=3$'

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'Find the polar coordinates of the center and the radius of the circle represented by the following polar equations:'

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Join our Discord#### Q.49

'For the cubic function $f(x)=x^{3}+bx+c$, find all linear functions $g(x)$ that satisfy $g(f(x))=f(g(x))$.'

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'When a function \ f \ maps a set \ A \ to a set \ B \, we call the set \ A \ the domain.'

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'Using mathematical induction, prove that this equation holds for all natural numbers n.'

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'Let P be the moving point on the curve xy=4. A perpendicular line PQ is drawn from P to the y-axis such that Q moves along the y-axis at a speed of 2 units per second. Find the velocity and acceleration of P when it passes through the point (2,2).'

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'The inverse function of a function, f^{-1}(x) = f(x)'

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'(1) Summary\n(1) S(a)=\\frac{1}{2}\\sqrt{5a^{2}+6a+90}=\\frac{1}{2}\\sqrt{5\\left(a+\\frac{3}{5}\\right)^{2}+\\frac{441}{5}}\nTherefore, S(a) takes the minimum value of \\frac{1}{2}\\sqrt{\\frac{441}{5}}=\\frac{21\\sqrt{5}}{10} when a=-\\frac{3}{5}.'

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'Find the equations of the tangent and normal lines at point P on the following curve.'

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'Two curves with perpendicular tangents at a common point. The two curves, y=x^{2}+a x+b and y=\\frac{c}{x}, intersect at the point (2,1) and the tangents at this point are perpendicular to each other. Find the values of the constants a, b, and c.'

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'\\n(1)\\\ y^{\\prime}=4 x^{3}-2 \\cdot 3 x^{2}+3 \\cdot 1-0=4 x^{3}-6 x^{2}+3 \'

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Join our Discord#### Q.58

'Find the following definite integral.\\[ \\int_{a}^{b}(x-a)^{2}(x-b)^{2} \\,dx \\]'

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'The graph of the function y=√(1+x^2) passes through the two points A(0,1) and B(1,√2). The equation of the line AB is y=(√2-1)x+1. In the range 0≤x≤1, 1≤√(1+x^2)≤(√2-1)x+1 always holds, and the equality usually does not hold.'

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'The functions f(x) and g(x) are continuous on the interval [a, b], with the maximum value of f(x) greater than the maximum value of g(x), and the minimum value of f(x) less than the minimum value of g(x). Show that the equation f(x)=g(x) has a real number solution in the range a ≤ x ≤ b.'

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Join our Discord#### Q.62

'Find the tangent and normal lines on the curve y^2=4px.'

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Join our Discord#### Q.64

'Common points on the graphs of a function and its inverse function'

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Join our Discord#### Q.65

'Practice: Find the value of the constant a when the sum of the local maximum and local minimum of the function f(x)=2x^3+ax^2+(a-4)x+2 is 6.'

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Join our Discord#### Q.67

"Use a, f(a), and f'(a) to express the remainder when the polynomial f(x) is divided by (x-a)^{2}."

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Join our Discord#### Q.68

'Find the maximum and minimum values of the function y = x^3 - 3x + 1.'

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Join our Discord#### Q.69

'Determine the values of constants a and b to satisfy the following conditions:'

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Join our Discord#### Q.71

'Complete the table to show the increase and decrease of the function y=-x^3+9x, and find the extremum and its points'

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Join our Discord#### Q.72

'Given the curve C: y = x^{3} + 3x^{2} + x and a point A(1, a). If there are 3 tangents that can be drawn through A and touch C, find the range of values for the constant a.'

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Join our Discord#### Q.73

'Find the values of the constants a and b for the function f(x)=x^{3}-a x^{2}+b such that the maximum value is 5 and the minimum value is 1.'

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Join our Discord#### Q.74

'Let a be a real number, and let curve C be y=x^3+(a-4)x^2+(-4a+2)x-2.'

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Join our Discord#### Q.76

'For the parabola y = 2x^2 + a and the circle x^2 + (y - 2) ^2 = 1, find the following: (1) The value of the constant a when the parabola and the circle are tangent (2) The range of values of the constant a that have four distinct intersection points'

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Join our Discord#### Q.77

'When the interval is $-\\frac{1}{2} \\leqq t \\leqq 2$, find the maximum and minimum values of $F(t)=\\int_{0}^{1} x|x-t| d x$.'

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'Prove that the midpoint M of the line segment connecting the two points (α, f(α)) and (β, f(β)) lies on the curve y=f(x), where the function y=f(x) has a maximum at x=α and a minimum at x=β.'

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Join our Discord#### Q.79

'Find the maximum and minimum values of the given function. Also, determine the corresponding values of x.'

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Join our Discord#### Q.80

'The given problem is about finding conditions and illustrating the region where the points satisfying those conditions exist.'

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Join our Discord#### Q.81

'Practice finding the following definite integrals.'

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Join our Discord#### Q.82

'Find the values of constants a, b, and c when the curves y=x^{3}+a x and y=b x^{2}+c both pass through the point (-1,0) and have a common tangent at that point. Also, determine the equation of the common tangent at that point.'

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Join our Discord#### Q.85

'Show the condition in which the equations g(x)=0 and h(x)=0 have two different real solutions each and no common solutions. Here g(x)=f(x)-x=a x^{2}-x-b, h(x)=a f(x)+a x+1=a^{2} x^{2}+a x-a b+1, and when f(f(x))-x=0, g(x) h(x)=0 has four different real solutions.'

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Join our Discord#### Q.86

'When the function $f(x)=4x^3-3(2a+1)x^2+6ax$ has local maxima and minima, find the condition that the constant $a$ must satisfy.'

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Join our Discord#### Q.87

'Consider Basic Exercise 120. Examine the sign changes of the following polynomials $f(x, y)$ and determine the signs in each region.'

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'What is the Chinese translation of the given text?'

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Join our Discord#### Q.91

'Practice: Draw the graph of the function y=| -x^3 + 9x |.'

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Join our Discord#### Q.92

'Find the range of values for the constant k when the function f(x)=x^{4}-8x^{3}+18kx^{2} does not have a local maximum.'

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Join our Discord#### Q.93

'When the point P(x, y) moves along the unit circle in the plane, find the maximum value of 15x^2 + 10xy - 9y^2 and the coordinates of point P that give the maximum value.'

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Join our Discord#### Q.94

'Let a be a positive constant. Find the minimum value of the function f(x) on the interval 0 ≤ x ≤ 2. Where: f(x) = -\\frac{x^{3}}{3} + \\frac{3}{2}ax^{2} - 2a^{2}x + a^{3}'

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Join our Discord#### Q.95

'Points to consider when drawing the graph of a cubic function'

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Join our Discord#### Q.96

'Practice plotting the graph of the following functions.'

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Join our Discord#### Q.97

'Find all functions f(x) that take extreme values when x = 1 and x = 3, for the function f(x) = ax^3 + bx^2 + cx + d. Also, provide the maximum and minimum values.'

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Join our Discord#### Q.98

'Find the value of the constant a when the curves y=x^{3}-x^{2}-12 x-1, y=-x^{3}+2 x^{2}+a are tangent. Also, find the equation of the tangent line at that point.'

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Join our Discord#### Q.99

'Find the range of values for the constant a such that the function f(x) = 2x^3 + ax^2 + ax + 1 always increases monotonically.'

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'Find the maximum and minimum values of the function. Also, determine the corresponding values of x. (219 (1) y=-x^{3}+12x+15 (-3 ≤ x ≤ 5))'

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'The number of real solutions of the equation f(t)=b is determined by the number of intersections between the graph of y=f(t) and the line y=b: 1 intersection when b<2a and b>e^{a}-e^{-a}; 2 intersections when b=2a or b=e^{a}-e^{-a}; 3 intersections when 2a<b<e^{a}-e^{-a}'

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'Translate the given text into multiple languages.'

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'Using the functions f(x)=x^{2}+1 and g(x)=2x-1, find the composite function (g ∘ f)(x).'

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'Plot the graphs of the following functions and find their ranges:\n(1) y=\\sqrt{3 x-4}\n(2) y=\\sqrt{-2 x+4}(-2 \\leqq x \\leqq 1)\n(3) y=\\sqrt{2-x}-1'

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'Determine the values of the constants a and b, such that the function y = √(4-x) takes values between 1 and 2 for a ≤ x ≤ b.'

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'Please sketch the graph of the function $y$ determined by the equation $y^{2}=x^{2}(8-x^{2})$ with respect to $x$.'

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'A function f(x) is called a continuous function when it is continuous for all values of x in its domain. Functions represented by polynomials, fractions, irrational functions, trigonometric functions, exponential functions, and logarithmic functions are all continuous functions.\nPlease prove that the function f(x)=√x is continuous in its domain.'

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"Proof that a quadratic function g(x) satisfying f(a)=g(a), f'(a)=g'(a), and f''(a)=g''(a) is g(x)=f(a)+f'(a)(x-a)+(f''(a)/2)(x-a)^2 (1)."

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'To find the value of the constant a such that the line y=ax+1 is tangent to the curve y=sqrt(2x-5)-1, the number of real solutions of the equation sqrt(2x-5)-1=ax+1 needs to be determined. However, treat double roots as one.'

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'1) The minimum value when t=2 is -\\frac{512 \\sqrt{2}}{15}'

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'When a point z or the origin O moves along a circle with a radius of 1 centered at the origin, what kind of figure does the point w, represented by the following equation, form? (3) w=2z-frac{2}{z}'

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'When a point (x, y) on the coordinate plane moves on the set defined by (x² + y²)² - (3x² - y²)y = 0, x ≥ 0, y ≥ 0, find the maximum value of x² + y² and the values of x, y that give this maximum value.'

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'Please investigate the function y = x sqrt(x + 1) where x > -1.'

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'For real number x, [x] represents the integer n satisfying n ≤ x < n+1, determine the values of constants a and b such that the function f(x) = ([x]+a)(b x-[x]) is continuous at x = 1 and x = 2.'

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'(1) Represent I as a function of a. (2) Find the minimum value of I and the corresponding value of a.'

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'A point P moves along a number line starting from the origin, and its coordinates as a function of time t are given by x = t^3 - 10t^2 + 24t (t > 0). Determine the velocity v and acceleration alpha of point P when it returns to the origin.'

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'Find the maximum and minimum values of x^2 - y^2 + xy when real numbers x and y satisfy 2x^2 + 3y^2 = 1.'

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'(6) At x=-4/5, the maximum value is 12√[3]{10}/25, at x=0, the minimum value is 0'

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'Find the minimum distance between a point $P$ on the ellipse $\\frac{x^{2}}{4}+y^{2}=1$ and the fixed point $A(a, 0)$. Here, $a$ is a real constant.'

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'Find the area S of the region enclosed by the curve and the x-axis for the following functions: (1) y=-x^4+2x^3, (2) y=x+4/x-5, (3) y=10-9e^{-x}-e^x'

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'When a point (x, y) on the coordinate plane moves on the set defined by (x^2 + y^2)^2 - (3x^2 - y^2)y = 0, x ≥ 0, y ≥ 0, find the maximum value of x^2 + y^2 and the values of x, y that give this maximum value.'

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'Find the range of x for Chapter 1 Functions - PRACTICE, problem 4.'

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'Please sketch the rough graph of the function of x determined by the equation y^2 = x^2(x+1) (convexity/concavity does not need to be considered).'

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'Find the point of Exercise 11 in Chapter 1 Functions.'

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'Question 66\n(1) Find the solution to the equation (x-3)² + y² + (z-2)² = 13.\n(2) Find the solution to the equation (x-2)² + (y-4)² + (z+1)² = 27.\n(3) Find the solution to the equation (x-2)² + (y+3)² + (z-1)² = 9.'

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'(2) Point B is a point obtained by rotating point A about the origin O by π/4 or -π/4 and increasing the distance from the origin by a factor of √2.'

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'For a point P moving on the coordinate plane, if the coordinates of P are described by the following equations as a function of time t, find the speed and magnitude of acceleration at t=1:\n(1) x=t^{2}, y=2 t\n(2) x=t, y=e^{-2 t}'

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'The position x of a point P moving along a line at time t is given by x=-2t^3+3t^2+8(t≥0). Find the velocity and acceleration of P when it is furthest from the origin O in the positive direction.'

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Join our Discord#### Q.34

'Plot the graphs of the following functions. Also, find their domains and ranges.'

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'Investigate whether the function f(x) is continuous or discontinuous at x=0. Where [x] denotes the greatest integer not exceeding the real number x.'

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'Given sets A and B, when an element of A is determined, the corresponding element of B is also determined as one. This correspondence is called a mapping from A to B. Mappings are denoted by symbols like f, g. It is denoted as f: A→B, to represent a mapping from A to B. For a mapping f from A to B, the element of B corresponding to an element a of A is called the image of a under f, denoted by f(a). For example, let A={a, b, c, d}, B={1, 2, 3, 4}. If f(a)=f(b)=1, f(c)=3, f(d)=2, then f is a mapping from A to B.'

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'Find the equation of the tangent line at the given point on the following ellipses and hyperbolas.'

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'Calculate the area using a rotation transformation'

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Join our Discord#### Q.40

'Sketch the outline of the graph of the function y determined by the following equations (also examine concavity and convexity):'

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'At time t = 2, the acceleration of P, α, where α = dv/dt = 6t, needs to be determined.'

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Join our Discord#### Q.42

'Please solve the problem related to function equations.'

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Join our Discord#### Q.43

'Please find the number of shared points on the graphs of two functions.'

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Join our Discord#### Q.44

'Find the maximum and minimum values of the function y=(x^2+2x)^2-4(x^2+2x)-4 when -2<=x<=1.'

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Join our Discord#### Q.45

'Find the minimum value of the function y=x^4-6x^2+10.'

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'\\[f(f(x))=2 f(x)-1=2 \\cdot(2 x-1)-1=4 x-3\\]\nTherefore, the graph of \\( y=f(f(x)) \\) is as shown in Figure (2).'

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'For all real numbers x1 and x2 satisfying 0 ≤ x ≤ 4, the condition for f(x1) < g(x2) to hold is that the maximum value of f(x) is less than the minimum value of g(x) for 0 ≤ x ≤ 4. Therefore, -a^2 + 8 < -3a - 10. Simplifying, we get a^2 - 3a - 18 > 0. Hence, (a + 3)(a - 6) > 0. Therefore, a < -3, 6 < a.'

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Join our Discord#### Q.48

'Plot the graph of the following functions: (1) y=x^{2}-4|x-1| (2) y=\\left|\\frac{1}{3} x^{2}+2 x-9\\right|'

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Join our Discord#### Q.49

"(2)'s graph is as shown on the right. To find the x-coordinate of the intersection of (1) and (2): [1] When x<-1, eliminate y from (1) and y=-2x+1 to get -x^{2}+2x+8=-2x+1, so x^{2}-4x-7=0, hence x=2±sqrt{11}, the values satisfying x<-1 are x=2-sqrt{11}; [2] When x≥2, eliminate y from (1) and y=2x-1 to get -x^{2}+2x+8=2x-1, so x^{2}=9, therefore the boundary is the x value where the expression in the || becomes 0, thus x+1=0 and x-2=0, giving x=-1 and x=2. Don't forget to check the conditions of this branch. [3] is the same."

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'For some real numbers x1, x2 satisfying 0 ≤ x ≤ 4, the condition for f(x1) < g(x2) to hold is that the minimum value of f(x) < the maximum value of g(x) holds for 0 ≤ x ≤ 4. Therefore, -a^2 - 1 < -3a - 1 can be simplified to a^2 - 3a > 0.'

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Find the equation of the hyperbola of 100th degree.
(1) rac{x^{2}}{9}-rac{y^{2}}{9}=1
(2) rac{x^{2}}{8}-rac{y^{2}}{18}=-1

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For the coordinates \( (x, y) \) of a point on the curve $C$, which are expressed using the variable $t$ as \( \left\{egin{array}{ll}x=t \\ y=t^{2}\end{array} \cdots \cdots \cdots(A)\right. \), investigate the values of $x$ and $y$ corresponding to the value of $t$, plot the points on the coordinate plane, and connect them with a smooth line. What kind of curve will be obtained?

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What kind of figure is represented by all points $z$ satisfying the following equations.
(1) $|2 z-1+2 i|=6$
(2) $|z+3 i|=|z+1|$

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When the point EX point \( \mathrm{P}(x, y) \) moves on the ellipse rac{x^{2}}{4}+y^{2}=1 , find the maximum and minimum values of $3 x^{2}-16 x y-12 y^{2}$.[Reference: Fukushima Medical University]

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When the point \( \mathrm{P}(x, y) \) moves on the ellipse $\frac{x^2}{4}+y^2=1$, find the maximum and minimum values of $3x^2-16xy-12y^2$.

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Express the curves represented by the following equations in polar coordinates. (a) 2x^2 + y^2 = 3 (b) y = x (c) x^2 + (y - 1)^2 = 1