The graph of the continuous function $g$, the derivative of the function $f$, is shown above. The function $g$ is piecewise linear for $- 5 \leq x < 3$, and $g ( x ) = 2 ( x - 4 ) ^ { 2 }$ for $3 \leq x \leq 6$.
(a) If $f ( 1 ) = 3$, what is the value of $f ( - 5 )$ ?
(b) Evaluate $\int _ { 1 } ^ { 6 } g ( x ) d x$.
(c) For $- 5 < x < 6$, on what open intervals, if any, is the graph of $f$ both increasing and concave up? Give a reason for your answer.
(d) Find the $x$-coordinate of each point of inflection of the graph of $f$. Give a reason for your answer.

Let $f$ be a continuous function defined on the closed interval $- 4 \leq x \leq 6$. The graph of $f$, consisting of four line segments, is shown above. Let $G$ be the function defined by $G ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
(a) On what open intervals is the graph of $G$ concave up? Give a reason for your answer.
(b) Let $P$ be the function defined by $P ( x ) = G ( x ) \cdot f ( x )$. Find $P ^ { \prime } ( 3 )$.
(c) Find $\lim _ { x \rightarrow 2 } \frac { G ( x ) } { x ^ { 2 } - 2 x }$.
(d) Find the average rate of change of $G$ on the interval $[ - 4,2 ]$. Does the Mean Value Theorem guarantee a value $c , - 4 < c < 2$, for which $G ^ { \prime } ( c )$ is equal to this average rate of change? Justify your answer.

The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$.
We admit that the second derivative of the function $f$ is defined on $\mathbb { R }$ by: $$f ^ { \prime \prime } ( x ) = ( 10 x + 25 ) \mathrm { e } ^ { x }$$ We can assert that: a. The function $f$ is convex on $\mathbb { R }$ b. The function $f$ is concave on $\mathbb { R }$ c. Point C is the unique inflection point of $\mathscr { C } _ { f }$ d. $\mathscr { C } _ { f }$ has no inflection point

Consider a function $f$ defined and twice differentiable on $\mathbb { R }$. We call $\mathscr { C }$ its graphical representation. We denote by $f ^ { \prime \prime }$ the second derivative of $f$. The curve of $f ^ { \prime \prime }$, denoted $\mathscr { C } ^ { \prime \prime }$, is represented in the graph opposite. a. $\mathscr { C }$ admits a unique inflection point; b. $f$ is convex on the interval $[ - 1 ; 2 ]$; c. $f$ is convex on $] - \infty ; - 1 ]$ and on $[2; + \infty [$; d. $f$ is convex on $\mathbb { R }$.

The purpose of this exercise is to study the function $f$ defined on the interval $]0; +\infty[$ by:
$$f(x) = x \ln(x^2) - \frac{1}{x}$$
Part A: graphical readings
Below is the representative curve $(\mathscr{C}_f)$ of the function $f$, as well as the line $(T)$, tangent to the curve $(\mathscr{C}_f)$ at point A with coordinates $(1; -1)$. This tangent also passes through the point $B(0; -4)$.

1. Read graphically $f'(1)$ and give the reduced equation of the tangent $(T)$.
2. Give the intervals on which the function $f$ appears to be convex or concave. What does point A appear to represent for the curve $(\mathscr{C}_f)$?

Part B: analytical study

1. Determine, by justifying, the limit of $f$ at $+\infty$, then its limit at 0.
2. It is admitted that the function $f$ is twice differentiable on the interval $]0; +\infty[$. a. Determine $f'(x)$ for $x$ belonging to the interval $]0; +\infty[$. b. Show that for all $x$ belonging to the interval $]0; +\infty[$, $$f''(x) = \frac{2(x+1)(x-1)}{x^3}.$$
3. a. Study the convexity of the function $f$ on the interval $]0; +\infty[$. b. Study the variations of the function $f'$, then the sign of $f'(x)$ for $x$ belonging to the interval $]0; +\infty[$. Deduce the direction of variation of the function $f$ on the interval $]0; +\infty[$.
4. a. Show that the equation $f(x) = 0$ has a unique solution $\alpha$ on the interval $]0; +\infty[$. b. Give the value of $\alpha$ rounded to the nearest hundredth and show that $\alpha$ satisfies: $$\alpha^2 = \exp\left(\frac{1}{\alpha^2}\right)$$

Let $f$ be the function defined on $\mathbb { R }$ by
$$f ( x ) = x \mathrm { e } ^ { - 2 x } .$$
We admit that $f$ is twice differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ the derivative of the function $f$. We denote $C _ { f }$ the representative curve of $f$ in an orthonormal coordinate system of the plane.
For each of the following statements, specify whether it is true or false, then justify the answer given.
Any answer without justification will not be taken into account.
Statement 1. For all real $x$, we have $f ^ { \prime } ( x ) = ( - 2 x + 1 ) \mathrm { e } ^ { - 2 x }$.
Statement 2. The function $f$ is a solution on $\mathbb { R }$ of the differential equation:
$$y ^ { \prime } + 2 y = \mathrm { e } ^ { - 2 x }$$
Statement 3. The function $f$ is convex on $] - \infty ; 1 ]$.
Statement 4. The equation $f ( x ) = - 1$ admits a unique solution on $\mathbb { R }$.
Statement 5. The area of the region bounded by the curve $C _ { f }$, the $x$-axis and the lines with equations $x = 0$ and $x = 1$ is equal to $\frac { 1 } { 4 } - \frac { 3 \mathrm { e } ^ { - 2 } } { 4 }$.

(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

Study the convexity of the function $t \mapsto \ln(1 + \mathrm{e}^t)$.

Study the convexity of the function $t \mapsto \ln \left( 1 + \mathrm { e } ^ { t } \right)$.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a twice differentiable function such that $\frac { d ^ { 2 } f ( x ) } { d x ^ { 2 } }$ is positive for all $x \in \mathbb { R }$, and suppose $f ( 0 ) = 1 , f ( 1 ) = 4$. Which of the following is not a possible value of $f ( 2 )$ ?
(A) 7 .
(B) 8 .
(C) 9 .
(D) 10 .

Let $f$ be any function continuous on $[ a , b ]$ and twice differentiable on $( a , b )$. If all $x \in ( a , b ) , f ^ { \prime } ( x ) > 0$ and $f ^ { \prime \prime } ( x ) < 0$, then for any $c \in ( a , b ) , \frac { f ( c ) - f ( a ) } { f ( b ) - f ( c ) }$
(1) $\frac { b + a } { b - a }$
(2) 1
(3) $\frac { b - c } { c - a }$
(4) $\frac { c - a } { b - c }$

Let $g(x) = f(x) + f(1 - x)$ and $f ^ { \prime \prime } (x) > 0 , x \in (0,1)$. If $g$ is decreasing in the interval $(0 , \alpha)$ and increasing in the interval $(\alpha , 1)$, then $\tan ^ { - 1 } (2\alpha) + \tan ^ { - 1 } \left( \frac { 1 } { \alpha } \right) + \tan ^ { - 1 } \left( \frac { \alpha + 1 } { \alpha } \right)$ is equal to
(1) $\pi$
(2) $\frac { 5\pi } { 4 }$
(3) $\frac { 3\pi } { 4 }$
(4) $\frac { 3\pi } { 2 }$

Let $g : R \rightarrow R$ be a non constant twice differentiable such that $g ^ { \prime } \left( \frac { 1 } { 2 } \right) = g ^ { \prime } \left( \frac { 3 } { 2 } \right)$. If a real valued function $f$ is defined as $f ( x ) = \frac { 1 } { 2 } [ g ( x ) + g ( 2 - x ) ]$, then
(1) $f ^ { \prime \prime } ( x ) = 0$ for atleast two $x$ in $( 0,2 )$
(2) $f ^ { \prime \prime } ( x ) = 0$ for exactly one $x$ in $( 0,1 )$
(3) $f ^ { \prime \prime } ( x ) = 0$ for no $x$ in $( 0,1 )$
(4) $f ^ { \prime } \left( \frac { 3 } { 2 } \right) + f ^ { \prime } \left( \frac { 1 } { 2 } \right) = 1$