We consider the function $f$ defined on the interval $]2; +\infty[$ by $$f(x) = x\ln(x-2)$$ Part of the representative curve $\mathscr{C}_f$ of the function $f$ is given below.

1. Conjecture, using the graph, the direction of variation of $f$, its limits at the boundaries of its domain of definition, and any possible asymptotes.
2. Solve the equation $f(x) = 0$ on $]2; +\infty[$.
3. Calculate $\displaystyle\lim_{\substack{x \rightarrow 2 \\ x > 2}} f(x)$. Does this result confirm one of the conjectures made in question 1?
4. Prove that for all $x$ belonging to $]2; +\infty[$: $$f'(x) = \ln(x-2) + \frac{x}{x-2}$$
5. We consider the function $g$ defined on the interval $]2; +\infty[$ by $g(x) = f'(x)$. a. Prove that for all $x$ belonging to $]2; +\infty[$, we have: $$g'(x) = \frac{x-4}{(x-2)^2}$$ b. We admit that $\displaystyle\lim_{\substack{x \rightarrow 2 \\ x > 2}} g(x) = +\infty$ and that $\displaystyle\lim_{x \rightarrow +\infty} g(x) = +\infty$. Deduce the table of variations of the function $g$ on $]2; +\infty[$. The exact value of the extremum of the function $g$ should be shown. c. Deduce that, for all $x$ belonging to $]2; +\infty[$, $g(x) > 0$. d. Deduce the direction of variation of the function $f$ on $]2; +\infty[$.
6. Study the convexity of the function $f$ on $]2; +\infty[$ and specify the coordinates of any possible inflection point of the representative curve of the function $f$.
7. How many values of $x$ exist for which the representative curve of $f$ admits a tangent with slope equal to 3?

We propose to study in this part the function $f$ encountered in Part B question 2. We recall that, for every real $x$, $f(x) = \left(6x^2 + 2x - 2\right)\mathrm{e}^{-5x+1}$. We denote $f'$ the derivative function of the function $f$. We call $\mathscr{C}_f$ the representative curve of $f$ in a coordinate system of the plane.

1. We admit that $\lim_{x \rightarrow +\infty} f(x) = 0$. Determine the limit of the function $f$ at $-\infty$.
2. Using Part A (Exercise 4), show that $\mathscr{C}_f$ intersects the $x$-axis at two points (the coordinates of these points are not expected).
3. Using Parts A and B (Exercise 4), show that $\mathscr{C}_f$ has two horizontal tangent lines.
4. Draw the complete variation table of the function $f$.
5. Determine by justifying the number of solution(s) of the equation $f(x) = 1$.
6. For every real $m$ strictly greater than 0.2, we define $I_m$ by $I_m = \int_{0.2}^{m} f(x)\,\mathrm{d}x$. a. Verify that the function $F$ defined on $\mathbb{R}$ by $$F(x) = \left(-\frac{6}{5}x^2 - \frac{22}{25}x + \frac{28}{125}\right)\mathrm{e}^{-5x+1}$$ is a primitive of the function $f$ on $\mathbb{R}$. b. Does there exist a value of $m$ for which $I_m = 0$? Interpret this result graphically.

We consider the function $f$ defined on the interval $] 0 ; + \infty [$ by $$f ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } } { 2 \sqrt { x } }$$ and we call $\mathscr { C } _ { f }$ its representative curve in an orthonormal coordinate system.

1. We define the function $g$ on the interval $] 0 ; + \infty \left[ \operatorname { by } g ( x ) = \mathrm { e } ^ { \sqrt { x } } \right.$. a. Show that $g ^ { \prime } ( x ) = f ( x )$ for all $x$ in the interval $] 0 ; + \infty [$. b. For all real $x$ in the interval $] 0 ; + \infty \left[ \right.$, calculate $f ^ { \prime } ( x )$ and show that: $$f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( \sqrt { x } - 1 ) } { 4 x \sqrt { x } } .$$
2. a. Determine the limit of the function $f$ at 0. b. Interpret this result graphically.
3. a. Determine the limit of the function $f$ at $+ \infty$. b. Study the direction of variation of the function $f$ on $] 0 ; + \infty [$. Draw the variation table of the function $f$ showing the limits at the boundaries of the domain of definition. c. Show that the equation $f ( x ) = 2$ has a unique solution on the interval $\left[ 1 ; + \infty \left[ \right. \right.$ and give an approximate value to $10 ^ { - 1 }$ of this solution.
4. We set $I = \int _ { 1 } ^ { 2 } f ( x ) \mathrm { d } x$. a. Calculate $I$. b. Interpret the result graphically.
5. We admit that the function $f$ is twice differentiable on the interval $] 0 ; + \infty [$ and that: $$f ^ { \prime \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( x - 3 \sqrt { x } + 3 ) } { 8 x ^ { 2 } \sqrt { x } } .$$ a. By setting $X = \sqrt { x }$, show that $x - 3 \sqrt { x } + 3 > 0$ for all real $x$ in the interval $] 0 ; + \infty [$. b. Study the convexity of the function $f$ on the interval $] 0 ; + \infty [$.

Consider the function $f$ defined on the interval $] 0 ; + \infty [$ by: $$f ( x ) = \frac { \ln ( x ) } { x ^ { 2 } } + 1$$ We denote $\mathscr { C } _ { f }$ the representative curve of the function $f$ in an orthonormal coordinate system. It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$ and we denote $f ^ { \prime }$ its derivative function.

1. Determine the limits of the function $f$ at 0 and at $+ \infty$. Deduce the possible asymptotes to the curve $\mathscr { C } _ { f }$.
2. Show that, for all real $x$ in the interval $] 0 ; + \infty [$, we have: $$f ^ { \prime } ( x ) = \frac { 1 - 2 \ln ( x ) } { x ^ { 3 } }$$
3. Deduce the variation table of the function $f$ on the interval $] 0 ; + \infty [$.
4. a. Show that the equation $f ( x ) = 0$ has a unique solution, denoted $\alpha$, on the interval $] 0 ; + \infty [$. b. Give an interval for the real number $\alpha$ with amplitude 0.01. c. Deduce the sign of the function $f$ on the interval $] 0 ; + \infty [$.
5. Consider the function $g$ defined on the interval $] 0 ; + \infty [$ by: $$g ( x ) = \ln ( x )$$ We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in an orthonormal coordinate system with origin O. We consider a strictly positive real $x$ and the point M of the curve $\mathscr{C} _ { g }$ with abscissa $x$. We denote OM the distance between points O and M. a. Express the quantity $\mathrm { OM } ^ { 2 }$ as a function of the real $x$. b. Show that, when the real $x$ ranges over the interval $] 0 ; + \infty [$, the quantity $\mathrm { OM } ^ { 2 }$ admits a minimum at $\alpha$. c. The minimum value of the distance OM, when the real $x$ ranges over the interval $] 0 ; + \infty [$, is called the distance from point O to the curve $\mathscr { C } _ { g }$. We denote $d$ this distance. Express $d$ in terms of $\alpha$.

A derivada da função $f(x) = x^3 - 3x^2 + 2x - 1$ é
(A) $f'(x) = 3x^2 - 6x + 2$ (B) $f'(x) = 3x^2 - 3x + 2$ (C) $f'(x) = x^2 - 6x + 2$ (D) $f'(x) = 3x^2 + 6x + 2$ (E) $f'(x) = 3x^2 - 6x - 2$

QUESTION 164
The derivative of $f(x) = x^3 - 4x^2 + 5x - 2$ is
(A) $f'(x) = 3x^2 - 8x + 5$
(B) $f'(x) = 3x^2 - 4x + 5$
(C) $f'(x) = 3x^2 + 8x + 5$
(D) $f'(x) = x^2 - 8x + 5$
(E) $f'(x) = 3x^2 - 8x - 5$

The derivative of $f(x) = x^3 - 3x^2 + 2x$ at $x = 1$ is:
(A) $-2$
(B) $-1$
(C) $0$
(D) $1$
(E) $2$

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function, where $\mathbb{R}$ denotes the set of real numbers. Suppose that for all real numbers $x$ and $y$, the function $f$ satisfies
$$f'(x) - f'(y) \leq 3|x - y|$$
Answer the following questions. No credit will be given without full justification.
(a) Show that for all $x$ and $y$, we must have $\left|f(x) - f(y) - f'(y)(x - y)\right| \leq 1.5(x - y)^2$.
(b) Find the largest and smallest possible values for $f''(x)$ under the given conditions.

(Calculus) For the function $f ( x ) = x + \sin x$, define the function $g ( x )$ as $$g ( x ) = ( f \circ f ) ( x )$$ Which of the following in are correct? [3 points]
ㄱ. The graph of function $f ( x )$ is concave down on the open interval $( 0 , \pi )$. ㄴ. The function $g ( x )$ is increasing on the open interval $( 0 , \pi )$. ㄷ. There exists a real number $x$ in the open interval $( 0 , \pi )$ such that $g ^ { \prime } ( x ) = 1$.
(1) ᄀ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ᄂ, ᄃ
(5) ᄀ, ᄂ, ᄃ

As shown in the figure, there is an isosceles triangle ABC with AB as one side of length 4, $\overline { \mathrm { AC } } = \overline { \mathrm { BC } }$, and $\angle \mathrm { ACB } = \theta$. On the extension of segment AB, a point D is taken such that $\overline { \mathrm { AC } } = \overline { \mathrm { AD } }$, and a point P is taken such that $\overline { \mathrm { AC } } = \overline { \mathrm { AP } }$ and $\angle \mathrm { PAB } = 2 \theta$. Let $S ( \theta )$ be the area of triangle BDP. Find the value of $\lim _ { \theta \rightarrow + 0 } ( \theta \times S ( \theta ) )$. (Here, $0 < \theta < \frac { \pi } { 6 }$) [4 points]

Let P be a point on the curve $y = 2 e ^ { - x }$ at $\mathrm { P } \left( t , 2 e ^ { - t } \right)$ $(t > 0)$. Let A be the foot of the perpendicular from P to the $y$-axis, and let B be the point where the tangent line at P intersects the $y$-axis. What is the value of $t$ that maximizes the area of triangle APB? [4 points]
(1) 1
(2) $\frac { e } { 2 }$
(3) $\sqrt { 2 }$
(4) 2
(5) $e$

For a positive number $t$, the function $f ( x )$ defined on the interval $[ 1 , \infty )$ is $$f ( x ) = \begin{cases} \ln x & ( 1 \leq x < e ) \\ - t + \ln x & ( x \geq e ) \end{cases}$$ Among linear functions $g ( x )$ satisfying the following condition, let $h ( t )$ be the minimum value of the slope of the line $y = g ( x )$.
For all real numbers $x \geq 1$, $( x - e ) \{ g ( x ) - f ( x ) \} \geq 0$.
For a differentiable function $h ( t )$, a positive number $a$ satisfies $h ( a ) = \frac { 1 } { e + 2 }$. What is the value of $h ^ { \prime } \left( \frac { 1 } { 2 e } \right) \times h ^ { \prime } ( a )$? [4 points]
(1) $\frac { 1 } { ( e + 1 ) ^ { 2 } }$
(2) $\frac { 1 } { e ( e + 1 ) }$
(3) $\frac { 1 } { e ^ { 2 } }$
(4) $\frac { 1 } { ( e - 1 ) ( e + 1 ) }$
(5) $\frac { 1 } { e ( e - 1 ) }$

For the function $f ( x ) = x ^ { 4 } - 3 x ^ { 2 } + 8$, find the value of $f ^ { \prime } ( 2 )$. [3 points]

A point P moving on the coordinate plane has position $( x , y )$ at time $t \left( 0 < t < \frac { \pi } { 2 } \right)$ given by
$$x = t + \sin t \cos t , \quad y = \tan t$$
What is the minimum speed of point P for $0 < t < \frac { \pi } { 2 }$? [3 points]
(1) 1
(2) $\sqrt { 3 }$
(3) 2
(4) $2 \sqrt { 2 }$
(5) $2 \sqrt { 3 }$

For the function $$f ( x ) = \begin{cases} - x & ( x \leq 0 ) \\ x - 1 & ( 0 < x \leq 2 ) \\ 2 x - 3 & ( x > 2 ) \end{cases}$$ and a non-constant polynomial $p ( x )$, which of the following statements are correct? [4 points]
ㄱ. If the function $p ( x ) f ( x )$ is continuous on the entire set of real numbers, then $p ( 0 ) = 0$. ㄴ. If the function $p ( x ) f ( x )$ is differentiable on the entire set of real numbers, then $p ( 2 ) = 0$. ㄷ. If the function $p ( x ) \{ f ( x ) \} ^ { 2 }$ is differentiable on the entire set of real numbers, then $p ( x )$ is divisible by $x ^ { 2 } ( x - 2 ) ^ { 2 }$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a real number $t$, let the equation of the tangent line to the curve $y = e ^ { x }$ at the point $\left( t , e ^ { t } \right)$ be $y = f ( x )$. Let $g ( t )$ be the minimum value of the real number $k$ such that the function $y = | f ( x ) + k - \ln x |$ is differentiable on the entire set of positive real numbers. For two real numbers $a , b ( a < b )$, let $\int _ { a } ^ { b } g ( t ) d t = m$. Which of the following statements in the given options are correct? [4 points]
$\langle$Options$\rangle$
ㄱ. There exist two real numbers $a , b ( a < b )$ such that $m < 0$. ㄴ. If $g ( c ) = 0$ for a real number $c$, then $g ( - c ) = 0$. ㄷ. If $m$ is minimized when $a = \alpha , b = \beta ( \alpha < \beta )$, then
$$\frac { 1 + g ^ { \prime } ( \beta ) } { 1 + g ^ { \prime } ( \alpha ) } < - e ^ { 2 }$$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a positive real number $t$, let $f ( t )$ be the value of the real number $a$ such that the curve $y = t ^ { 3 } \ln ( x - t )$ meets the curve $y = 2 e ^ { x - a }$ at exactly one point. Find the value of $\left\{ f ^ { \prime } \left( \frac { 1 } { 3 } \right) \right\} ^ { 2 }$. [4 points]

For the function $f ( x ) = x ^ { 4 } + 3 x - 2$, what is the value of $f ^ { \prime } ( 2 )$? [3 points]
(1) 35
(2) 37
(3) 39
(4) 41
(5) 43

Two polynomial functions $f ( x )$ and $g ( x )$ satisfy $$\lim _ { x \rightarrow 0 } \frac { f ( x ) + g ( x ) } { x } = 3 , \quad \lim _ { x \rightarrow 0 } \frac { f ( x ) + 3 } { x g ( x ) } = 2$$ For the function $h ( x ) = f ( x ) g ( x )$, what is the value of $h ^ { \prime } ( 0 )$? [4 points]
(1) 27
(2) 30
(3) 33
(4) 36
(5) 39

As shown in the figure, in a right triangle ABC with $\overline { \mathrm { AB } } = 2$ and $\angle \mathrm {~B} = \frac { \pi } { 2 }$, let D and E be the points where the circle with center A and radius 1 meets the two segments $\mathrm { AB }$ and $\mathrm { AC }$ respectively. Let F be the trisection point of arc DE closer to point D, and let G be the point where line AF meets segment BC. Let $\angle \mathrm { BAG } = \theta$. Let $f ( \theta )$ be the area of the common part of the interior of triangle ABG and the exterior of sector ADF, and let $g ( \theta )$ be the area of sector AFE. Find the value of $40 \times \lim _ { \theta \rightarrow 0 + } \frac { f ( \theta ) } { g ( \theta ) }$. (where $0 < \theta < \frac { \pi } { 6 }$) [3 points]

A point P starts at time $t = 0$ and moves on a number line. At time $t$ ($t \geq 0$), its position $x$ is given by $$x = t^{3} - \frac{3}{2}t^{2} - 6t$$ What is the acceleration of point P at the time when its direction of motion changes after starting? [4 points]
(1) 6
(2) 9
(3) 12
(4) 15
(5) 18

21. Given the function $f ( x ) = \frac { a x } { ( x + r ) ^ { 2 } } ( a > 0 , r > 0 )$
(1) Find the domain of $f ( x )$ and discuss the monotonicity of $f ( x )$;
(2) If $\frac { a } { r } = 400$, find the extreme values of $f ( x )$ on $( 0 , + \infty )$.

(12 points)
Given the function $f(x) = e^x(e^x - a) - a^2x$.
(1) Discuss the monotonicity of $f(x)$;
(2) If $f(x) \geq 0$, find the range of values for $a$.

(12 points)
Let the function $f(x) = (1-x^2)e^x$.
(1) Discuss the monotonicity of $f(x)$.
(2) When $x \geq 0$, $f(x) \leq ax + 1$. Find the range of values of $a$.

Given the function $f ( x ) = \mathrm { e } ^ { x } - a ( x + 2 )$ .
(1) When $a = 1$ , discuss the monotonicity of $f ( x )$ ;
(2) If $f ( x )$ has two zeros, find the range of values for $a$ .