Exercise 2 (6 points)

We consider the function $f$ defined on $[ 0 ; + \infty [$ by
$$f ( x ) = 5 \mathrm { e } ^ { - x } - 3 \mathrm { e } ^ { - 2 x } + x - 3$$
We denote $\mathscr { C } _ { f }$ the graphical representation of the function $f$ and $\mathscr { D }$ the line with equation $y = x - 3$ in an orthogonal coordinate system of the plane.

Part A: Relative positions of $\mathscr { C } _ { f }$ and $\mathscr { D }$

Let $g$ be the function defined on the interval $[ 0 ; + \infty [$ by $g ( x ) = f ( x ) - ( x - 3 )$.

1. Justify that, for every real number $x$ in the interval $[ 0 ; + \infty [ , g ( x ) > 0$.
2. Do the curve $\mathscr { C } _ { f }$ and the line $\mathscr { D }$ have a common point? Justify.

Part B: Study of the function $\boldsymbol { g }$

We denote $M$ the point with abscissa $x$ of the curve $\mathscr { C } _ { f } , N$ the point with abscissa $x$ of the line $\mathscr { D }$ and we are interested in the evolution of the distance $M N$.

1. Justify that, for all $x$ in the interval $[ 0 ; + \infty [$, the distance $M N$ is equal to $g ( x )$.
2. We denote $g ^ { \prime }$ the derivative function of the function $g$ on the interval $[ 0 ; + \infty [$.

For all $x$ in the interval $\left[ 0 ; + \infty \left[ \right. \right.$, calculate $g ^ { \prime } ( x )$.
3. Show that the function $g$ has a maximum on the interval $[ 0 ; + \infty [$ which we will determine. Give a graphical interpretation of this.

Part C: Study of an area

We consider the function $\mathscr { A }$ defined on the interval $[ 0 ; + \infty [$ by
$$\mathscr { A } ( x ) = \int _ { 0 } ^ { x } [ f ( t ) - ( t - 3 ) ] \mathrm { d } t$$

1. Shade on the graph given in Appendix 1 (to be returned with your work) the region whose area is given by $\mathscr { A } ( 2 )$.
2. Justify that the function $\mathscr { A }$ is increasing on the interval $[ 0 ; + \infty [$.
3. For all positive real $x$, calculate $\mathscr { A } ( x )$.
4. Does there exist a value of $x$ such that $\mathscr { A } ( x ) = 2$ ?

We consider the function $g$ defined for all real $x$ by $g(x) = \mathrm{e}^{1-x}$. The representative curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ of the functions $f$ and $g$ respectively are drawn in a coordinate system. The purpose of this part is to study the relative position of these two curves.

1. After observing the graph, what conjecture can be made?
2. Justify that, for all real $x$ belonging to $]-\infty; 0]$, $f(x) < g(x)$.
3. In this question, we consider the interval $]0; +\infty[$. We set, for all strictly positive real $x$, $\Phi(x) = \ln x - x^{2} + x$. a. Show that, for all strictly positive real $x$, $$f(x) \leqslant g(x) \text{ is equivalent to } \Phi(x) \leqslant 0$$ It is admitted for the rest that $f(x) = g(x)$ is equivalent to $\Phi(x) = 0$. b. It is admitted that the function $\Phi$ is differentiable on $]0; +\infty[$. Draw the table of variations of the function $\Phi$. (The limits at 0 and $+\infty$ are not required.) c. Deduce that, for all strictly positive real $x$, $\Phi(x) \leqslant 0$.
4. a. Is the conjecture made in question 1 of Part B valid? b. Show that $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ have a unique common point, denoted $A$. c. Show that at this point $A$, these two curves have the same tangent line.

(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 $a$.

Let $f ( x ) = ( x + a ) \ln ( x + b )$. If $f ( x ) \geq 0$, then the minimum value of $a ^ { 2 } + b ^ { 2 }$ is
A. $\frac { 1 } { 8 }$
B. $\frac { 1 } { 4 }$
C. $\frac { 1 } { 2 }$
D. 1

We are given $f \in \mathcal{C}^1(\mathbb{R})$ such that $f'$ is $L$-Lipschitzian, with $L > 0$, $f$ is $\alpha$-convex with $\alpha > 0$, and $x_*$ denotes a minimizer of $f$. Deduce that for all $x, y \in \mathbb{R}$, denoting $\tilde{x} := x - \tau f'(x)$ and $\tilde{y} := y - \tau f'(y)$, we have $$|\tilde{x} - \tilde{y}|^2 \leq |x-y|^2(1 - \alpha\tau(2 - L\tau)).$$

We are given $f \in \mathcal{C}^1(\mathbb{R})$ such that $f'$ is $L$-Lipschitzian, with $L > 0$, $f$ is $\alpha$-convex with $\alpha > 0$, and $x_*$ denotes a minimizer of $f$. We suppose $0 < \tau < 2/L$. Show that $\left|x_n - x_*\right| \leq \rho^n \left|x_0 - x_*\right|$, where $\rho$ is a constant that we will specify, and such that $0 \leq \rho < 1$.

We are given $f \in \mathcal{C}^1(\mathbb{R})$, convex, admitting a minimizer $x_* \in \mathbb{R}$, with $f'$ being $L$-Lipschitzian, and $0 < \tau < 2/L$. Show that $0 \leq f(x) - f(x_*) \leq |x - x_*||f'(x)|$ for all $x \in \mathbb{R}$.

We are given $f \in \mathcal{C}^1(\mathbb{R})$, convex, admitting a minimizer $x_* \in \mathbb{R}$, with $f'$ being $L$-Lipschitzian, and $0 < \tau < 2/L$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := x_n - \tau f'(x_n)$. Show that for all $n \in \mathbb{N}$, assuming $x_0 \neq x_*$, $$f(x_{n+1}) \leq f(x_n) - \frac{\tau}{2}(2 - \tau L)\frac{\left|f(x_n) - f(x_*)\right|^2}{\left|x_0 - x_*\right|^2}$$ Hint: use question 2.c)