Consider the function $f ( x ) = a x + \frac { 1 } { x + 1 }$, where $a$ is a positive constant. Let $L =$ the largest value of $f ( x )$ and $S =$ the smallest value of $f ( x )$ for $x \in [ 0,1 ]$. Show that $L - S > \frac { 1 } { 12 }$ for any $a > 0$.

10. If a function $f ( x )$ defined on $\mathbb{R}$ satisfies $f ( 0 ) = - 1$, and its derivative $f ^ { \prime } ( x )$ satisfies $f ^ { \prime } ( x ) > k > 1$, then among the following conclusions, the one that must be wrong is
A. $f \left( \frac { 1 } { k } \right) < \frac { 1 } { k }$
B. $f \left( \frac { 1 } { k } \right) > \frac { 1 } { k - 1 }$
C. $f \left( \frac { 1 } { k - 1 } \right) < \frac { 1 } { k - 1 }$
D. $f \left( \frac { 1 } { k - 1 } \right) > \frac { k } { k - 1 }$

Section II (Non-Multiple Choice Questions, 100 points)

II. Fill-in-the-Blank Questions: This section contains 5 questions, each worth 4 points, for a total of 20 points. Write your answers in the corresponding positions on the answer sheet.

20. Given the function $f ( x ) = 4 x - x ^ { 4 } , x \in \mathbb{R}$.
(1) Find the monotonicity of $f ( x )$;
(2) Let $P$ be the intersection point of the curve $y = f ( x )$ and the positive $x$-axis. The tangent line to the curve at point $P$ is $y = g ( x )$. Prove that for any positive real number $x$, we have $f ( x ) \leq g ( x )$;
(3) If the equation $f ( x ) = a$ (where $a$ is a real number) has two positive real roots $x _ { 1 } , x _ { 2 }$ with $x _ { 1 } < x _ { 2 }$, prove that $x _ { 2 } - x _ { 1 } < - \frac { a } { 3 } + 4 ^ { \frac { 1 } { 3 } }$.

Let $x$ be a strictly positive real number, $\beta$ a real number such that $0 < \beta < 1$.
Prove that: $x ^ { \beta } \leqslant \beta x + 1 - \beta$.

Show that
$$\forall ( x , y ) \in ] 0,1 \left[ ^ { 2 } , \quad \frac { x ( 1 - x ) y ( 1 - y ) } { 1 - x y } \leqslant \frac { 5 \sqrt { 5 } - 11 } { 2 } \right.$$

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 $x_{n+1} := x_n - \tau f'(x_n)$. a) Show that for all $x, y \in \mathbb{R}$ $$f(y) \geq f(x) + f'(x)(y-x)$$ Hint: consider a Taylor expansion of the convexity inequality when $t \rightarrow 0^+$. b) Show that for all $x, y \in \mathbb{R}$ $$f(y) \leq f(x) + f'(x)(y-x) + \frac{L}{2}(y-x)^2$$ c) Establish that for all $n \in \mathbb{N}$ $$f(x_{n+1}) \leq f(x_n) - \frac{\tau}{2}(2 - \tau L)\left|f'(x_n)\right|^2$$ Deduce that the sequence $\left(f(x_n)\right)_{n \in \mathbb{N}}$ is decreasing.

17. Prove that $\sin \mathrm { x } + 2 \mathrm { x } \geq \frac { 3 \mathrm { x } \cdot ( \mathrm { x } + 1 ) } { \pi } \forall \mathrm { x } \in \left[ 0 , \frac { \pi } { 2 } \right]$. (Justify the inequality, if any used).
Sol. Let $\mathrm { f } ( \mathrm { x } ) = 3 \mathrm { x } ^ { 2 } + ( 3 - 2 \pi ) \mathrm { x } - \pi \sin \mathrm { x }$ $\mathrm { f } ( 0 ) = 0 , \mathrm { f } \left( \frac { \pi } { 2 } \right) = - \mathrm { ve }$ $\mathrm { f } ^ { \prime } ( \mathrm { x } ) = 6 \mathrm { x } + 3 - 2 \pi - \pi \cos \mathrm { x }$ $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } ) = 6 + \pi \sin \mathrm { x } > 0$ $\Rightarrow \mathrm { f } ^ { \prime } ( \mathrm { x } )$ is increasing function in $\left[ 0 , \frac { \pi } { 2 } \right]$ ⇒ there is no local maxima of $\mathrm { f } ( \mathrm { x } )$ in $\left[ 0 , \frac { \pi } { 2 } \right]$ ⇒ graph of $\mathrm { f } ( \mathrm { x } )$ always lies below the x -axis [Figure] in $\left[ 0 , \frac { \pi } { 2 } \right]$. $\Rightarrow \mathrm { f } ( \mathrm { x } ) \leq 0$ in $\mathrm { x } \in \left[ 0 , \frac { \pi } { 2 } \right]$. $3 \mathrm { x } ^ { 2 } + 3 \mathrm { x } \leq 2 \pi \mathrm { x } + \pi \sin \mathrm { x } \Rightarrow \sin \mathrm { x } + 2 \mathrm { x } \geq \frac { 3 \mathrm { x } ( \mathrm { x } + 1 ) } { \pi }$.

The least value of $\alpha \in \mathbb{R}$ for which $4\alpha x^2 + \frac{1}{x} \geq 1$, for all $x > 0$, is
(A) $\frac{1}{64}$
(B) $\frac{1}{32}$
(C) $\frac{1}{27}$
(D) $\frac{1}{25}$

Let $\psi _ { 1 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad \psi _ { 2 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad f : [ 0 , \infty ) \rightarrow \mathbb { R }$ and $g : [ 0 , \infty ) \rightarrow \mathbb { R }$ be functions such that $f ( 0 ) = g ( 0 ) = 0$, $$\begin{gathered} \psi _ { 1 } ( x ) = e ^ { - x } + x , \quad x \geq 0 \\ \psi _ { 2 } ( x ) = x ^ { 2 } - 2 x - 2 e ^ { - x } + 2 , \quad x \geq 0 \\ f ( x ) = \int _ { - x } ^ { x } \left( | t | - t ^ { 2 } \right) e ^ { - t ^ { 2 } } d t , \quad x > 0 \end{gathered}$$ and $$g ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t } e ^ { - t } d t , \quad x > 0$$ Which of the following statements is TRUE ?
(A) $f ( \sqrt { \ln 3 } ) + g ( \sqrt { \ln 3 } ) = \frac { 1 } { 3 }$
(B) For every $x > 1$, there exists an $\alpha \in ( 1 , x )$ such that $\psi _ { 1 } ( x ) = 1 + \alpha x$
(C) For every $x > 0$, there exists a $\beta \in ( 0 , x )$ such that $\psi _ { 2 } ( x ) = 2 x \left( \psi _ { 1 } ( \beta ) - 1 \right)$
(D) $f$ is an increasing function on the interval $\left[ 0 , \frac { 3 } { 2 } \right]$

Let $\psi _ { 1 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad \psi _ { 2 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad f : [ 0 , \infty ) \rightarrow \mathbb { R }$ and $g : [ 0 , \infty ) \rightarrow \mathbb { R }$ be functions such that $f ( 0 ) = g ( 0 ) = 0$, $$\begin{gathered} \psi _ { 1 } ( x ) = e ^ { - x } + x , \quad x \geq 0 \\ \psi _ { 2 } ( x ) = x ^ { 2 } - 2 x - 2 e ^ { - x } + 2 , \quad x \geq 0 \\ f ( x ) = \int _ { - x } ^ { x } \left( | t | - t ^ { 2 } \right) e ^ { - t ^ { 2 } } d t , \quad x > 0 \end{gathered}$$ and $$g ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t } e ^ { - t } d t , \quad x > 0$$ Which of the following statements is TRUE ?
(A) $\psi _ { 1 } ( x ) \leq 1$, for all $x > 0$
(B) $\psi _ { 2 } ( x ) \leq 0$, for all $x > 0$
(C) $f ( x ) \geq 1 - e ^ { - x ^ { 2 } } - \frac { 2 } { 3 } x ^ { 3 } + \frac { 2 } { 5 } x ^ { 5 }$, for all $x \in \left( 0 , \frac { 1 } { 2 } \right)$
(D) $g ( x ) \leq \frac { 2 } { 3 } x ^ { 3 } - \frac { 2 } { 5 } x ^ { 5 } + \frac { 1 } { 7 } x ^ { 7 }$, for all $x \in \left( 0 , \frac { 1 } { 2 } \right)$

Q73. If the function $f ( x ) = \left( \frac { 1 } { x } \right) ^ { 2 x } ; x > 0$ attains the maximum value at $x = \frac { 1 } { \mathrm { e } }$ then:
(1) $\mathrm { e } ^ { \pi } < \pi ^ { \mathrm { e } }$
(2) $\mathrm { e } ^ { \pi } > \pi ^ { \mathrm { e } }$
(3) $( 2 e ) ^ { \pi } > \pi ^ { ( 2 e ) }$
(4) $\mathrm { e } ^ { 2 \pi } < ( 2 \pi ) ^ { \mathrm { e } }$

The function $f(x) = mx - 1 + \frac{1}{x}$ is given.
Accordingly, what is the smallest value of $m$ that satisfies the property $f(x) \geq 0$ for all $x > 0$?
A) $\frac{1}{2}$
B) $\frac{1}{3}$
C) $\frac{1}{4}$
D) $\frac{1}{5}$
E) $\frac{1}{6}$