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.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Show that for all $x \in B(a,r)$, the linear map $df(x)$ is injective.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Let $y_0 \in E$ such that $\left\|y_0 - f(a)\right\| \leqslant \frac{r}{4}$, and let $x_0 \in B(a,r)$ be a point where the application $x \mapsto \left\|y_0 - f(x)\right\|^2$ attains its minimum on $\overline{B(a,r)}$.
Show that $f(x_0) = y_0$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Let $V$ be a convex open set of $E$ and $h$ a function of class $\mathscr{C}^1$ from $V$ to $E$. Suppose that there exists a real number $C \geqslant 0$ such that for all $x \in V$, $\|dh(x)\| \leqslant C$. Show that for all $x_1$ and $x_2$ in $V$, we have $\left\|h(x_2) - h(x_1)\right\| \leqslant C \left\|x_2 - x_1\right\|$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Show that there exists a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Show that for all $x \in B(a,r)$, the linear map $df(x)$ is injective.

Let $f , g$ be differentiable functions on the real line $\mathbb { R }$ with $f ( 0 ) > g ( 0 )$. Assume that the set $M = \{ t \in \mathbb { R } \mid f ( t ) = g ( t ) \}$ is non-empty and that $f ^ { \prime } ( t ) \geq g ^ { \prime } ( t )$ for all $t \in M$. Then which of the following is necessarily true?
(A) If $t \in M$, then $t < 0$.
(B) For any $t \in M , f ^ { \prime } ( t ) > g ^ { \prime } ( t )$.
(C) For any $t \notin M , f ( t ) > g ( t )$.
(D) None of the above.

Let $f ( x ) , g ( x )$ be functions on the real line $\mathbb { R }$ such that both $f ( x ) + g ( x )$ and $f ( x ) g ( x )$ are differentiable. Which of the following is FALSE ?
(A) $f ( x ) ^ { 2 } + g ( x ) ^ { 2 }$ is necessarily differentiable.
(B) $f ( x )$ is differentiable if and only if $g ( x )$ is differentiable.
(C) $f ( x )$ and $g ( x )$ are necessarily continuous.
(D) If $f ( x ) > g ( x )$ for all $x \in \mathbb { R }$, then $f ( x )$ is differentiable.

Let $g : ( 0 , \infty ) \rightarrow ( 0 , \infty )$ be a differentiable function whose derivative is continuous, and such that $g ( g ( x ) ) = x$ for all $x > 0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.

Let $\mathbb { R }$ denote the set of all real numbers. Define the function $f : \mathbb { R } \rightarrow \mathbb { R }$ by
$$f ( x ) = \left\{ \begin{array} { c c } 2 - 2 x ^ { 2 } - x ^ { 2 } \sin \frac { 1 } { x } & \text { if } x \neq 0 \\ 2 & \text { if } x = 0 \end{array} \right.$$
Then which one of the following statements is TRUE?

(A)	The function $f$ is NOT differentiable at $x = 0$
(B)	There is a positive real number $\delta$, such that $f$ is a decreasing function on the interval ( $0 , \delta$ )
(C)	For any positive real number $\delta$, the function $f$ is NOT an increasing function on the interval ( $- \delta , 0$ )
(D)	$x = 0$ is a point of local minima of $f$

A value of $C$ for which the conclusion of Mean Value Theorem holds for the function $f ( x ) = \log _ { \mathrm { e } } x$ on the interval $[ 1,3 ]$ is
(1) $2 \log _ { 3 } e$
(2) $\frac { 1 } { 2 } \log _ { e } 3$
(3) $\log _ { 3 } e$
(4) $\log _ { e } 3$

Let the function, $f : [-7, 0] \rightarrow R$ be continuous on $[-7, 0]$ and differentiable on $(-7, 0)$. If $f(-7) = -3$ and $f ^ { \prime } (x) \leq 2$ for all $x \in (-7, 0)$, then for all such functions $f$, $f(-1) + f(0)$ lies in the interval
(1) $(-\infty, 20]$
(2) $[-3, 11]$
(3) $(-\infty, 11]$
(4) $[-6, 20]$

The value of $c$, in the Lagrange's mean value theorem for the function $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 8 x + 11$, when $x \in [ 0,1 ]$, is
(1) $\frac { 4 - \sqrt { 5 } } { 3 }$
(2) $\frac { 4 - \sqrt { 7 } } { 3 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { \sqrt { 7 } - 2 } { 3 }$

Let $f : ( 0 , \infty ) \rightarrow ( 0 , \infty )$ be a differentiable function such that $f ( 1 ) = e$ and $\lim _ { t \rightarrow x } \frac { t ^ { 2 } f ^ { 2 } ( x ) - x ^ { 2 } f ^ { 2 } ( t ) } { t - x } = 0$. If $f ( x ) = 1$, then $x$ is equal to:
(1) $\frac { 1 } { e }$
(2) $2 e$
(3) $\frac { 1 } { 2 e }$
(4) $e$

Let $f : S \rightarrow S$ where $S = ( 0 , \infty )$ be a twice differentiable function such that $f ( x + 1 ) = x f ( x )$. If $g : S \rightarrow R$ be defined as $g ( x ) = \log _ { \mathrm { e } } f ( x )$, then the value of $\left| g ^ { \prime \prime } ( 5 ) - g ^ { \prime \prime } ( 1 ) \right|$ is equal to :
(1) $\frac { 205 } { 144 }$
(2) $\frac { 197 } { 144 }$
(3) $\frac { 187 } { 144 }$
(4) 1