Let the function $f ( x ) = \frac { x ^ { 2 } } { 2 } - k \ln x$, $k > 0$\n(I) Find the monotonic intervals and extreme values of $f ( x )$;\n(II) Prove that if $f ( x )$ has a zero point, then $f ( x )$ has exactly one zero point on the interval $( 1 , \sqrt { e } )$.

Given the function $\mathrm { f } ( \mathrm { x } ) = \mathrm { a } x ^ { 3 } + x ^ { 2 } ( \mathrm { a } \in \mathrm { R } )$ has an extremum at $\mathrm { x } = - \frac { 4 } { 3 }$ .
(I) Determine the value of $a$;
(II) Let $\mathrm { g } ( \mathrm { x } ) = \mathrm { f } ( \mathrm { x } ) e ^ { x }$. Discuss the monotonicity of $\mathrm { g } ( \mathrm { x } )$.

19. Given the function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b ( a , b \in R )$.
(1) Discuss the monotonicity of $f ( x )$;
(2) If $b = c - a$ (where the real number c is a constant independent of a), when the function $f ( x )$ has three distinct zeros, the range of a is exactly $( - \infty , - 3 ) \cup \left( 1 , \frac { 3 } { 2 } \right) \cup \left( \frac { 3 } { 2 } , + \infty \right)$, find the value of c.

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 $x _ { 2 } - x _ { 1 } < - \frac { a } { 3 } + 4 ^ { \frac { 1 } { 3 } }$.

21. (This question is worth 14 points) Given the function $f ( x ) = - 2 \ln x + x ^ { 2 } - 2 a x + a ^ { 2 }$, where $a > 0$. (1) Let $g ( x )$ be the derivative of $f ( x )$. Discuss the monotonicity of $g ( x )$; (2) Prove: there exists $a \in ( 0,1 )$ such that $f ( x ) \geq 0$ holds for all $x$

21. Given the function $f ( x ) = - 2 ( x + a ) \ln x + x ^ { 2 } - 2 a x - 2 a ^ { 2 } + a$, where $a > 0$.
(1) Let $g ( x )$ be the derivative of $f ( x )$. Discuss the monotonicity of $g ( x )$;
(2) Prove: there exists $a \in ( 0, 1 )$ such that $f ( x ) \geq 0$ holds on the interval $(1, + \infty)$, and $f ( x ) = 0$ has a unique solution in $(1, + \infty)$.

Given the function $f ( x ) = \frac { 1 } { 3 } x ^ { 3 } - a \left( x ^ { 2 } + x + 1 \right)$.
(1) When $a = 3$, find the monotonic intervals of $f ( x )$;
(2) Prove: $f ( x )$ has exactly one zero.

Given the function $f ( x ) = \frac { 1 } { 4 } x ^ { 3 } - x ^ { 2 } + x$. (I) Find the equation of the tangent line to the curve $y = f ( x )$ with slope 1; (II) When $x \in [ - 2,4 ]$, prove that: $x - 6 \leqslant f ( x ) \leqslant x$; (III) Let $F ( x ) = | f ( x ) - ( x + a ) | ( a \in \mathbb { R } )$. Let $M ( a )$ denote the maximum value of $F ( x )$ on the interval $[ - 2,4 ]$. When $M ( a )$ is minimized, find the value of $a$.

A cone has a base radius of 1 and slant height of 3. The volume of the largest sphere that can be inscribed in this cone is $\_\_\_\_$ .

20. Let $f ( x ) = a ^ { 2 } x ^ { 2 } + a x - 3 \ln x + 1$, where $a > 0$.
(1) Discuss the monotonicity of $f ( x )$;
(2) If the graph of $y = f ( x )$ has no common points with the $x$-axis, find the range of values for $a$.

Given that sphere $O$ has radius $1$, and a quadrangular pyramid has vertex at $O$ with the four vertices of its base all on the surface of sphere $O$. When the volume of this quadrangular pyramid is maximum, its height is
A. $\frac{1}{3}$
B. $\frac{1}{2}$
C. $\frac{\sqrt{3}}{3}$
D. $\frac{\sqrt{2}}{2}$

10. Given the function $f ( x ) = x ^ { 3 } - x + 1$, then
A. $f ( x )$ has two extreme points
B. $f ( x )$ has three zeros
C. The point $( 0,1 )$ is a center of symmetry of the curve $y = f ( x )$
D. The line $y = 2 x$ is a tangent line to the curve $y = f ( x )$

The function $f ( x ) = \cos x + ( x + 1 ) \sin x + 1$ on the interval $[ 0,2 \pi ]$ has minimum and maximum values respectively
A. $- \frac { \pi } { 2 } , \frac { \pi } { 2 }$
B. $- \frac { 3 \pi } { 2 } , \frac { \pi } { 2 }$
C. $- \frac { \pi } { 2 } , \frac { \pi } { 2 } + 2$
D. $- \frac { 3 \pi } { 2 } , \frac { \pi } { 2 } + 2$

A sphere $O$ has radius 1. A pyramid has its apex at $O$ and the four vertices of its base all on the surface of sphere $O$. When the volume of this pyramid is maximized, its height is
A. $\frac { 1 } { 3 }$
B. $\frac { 1 } { 2 }$
C. $\frac { \sqrt { 3 } } { 3 }$
D. $\frac { \sqrt { 2 } } { 2 }$

Given that $x = x_1$ and $x = x_2$ are the local minimum and local maximum points respectively of the function $f(x) = 2a^x - ex^2$ ($a > 0$ and $a \neq 1$). If $x_1 < x_2$, then the range of $a$ is $\_\_\_\_$.

If the function $f(x)=a\ln x+\frac{b}{x}+\frac{c}{x^2}$ $(a\neq 0)$ has both a local maximum and a local minimum, then:
A. $bc>0$
B. $ab>0$
C. $b^2+8ac>0$
D. $ac<0$

Given $f(x) = ax - \frac{\sin x}{\cos^{2} x} , \quad x \in \left(0 , \frac{\pi}{2}\right)$ ,
(1) When $a = 8$ , discuss the monotonicity of $f(x)$ ;
(2) If $f(x) < \sin 2x$ , find the range of values for $a$ .

Given the function $f ( x ) = \left( \frac { 1 } { x } + a \right) \ln ( 1 + x )$.
(1) When $a = - 1$, find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 1 , f ( 1 ) )$.
(2) Do there exist $a$ and $b$ such that the curve $y = f \left( \frac { 1 } { x } \right)$ is symmetric about the line $x = b$? If they exist, find the values of $a$ and $b$. If they do not exist, explain why.
(3) If $f ( x )$ has an extremum on $( 0 , + \infty )$, find the range of $a$.

Given function $f ( x ) = \left\{ \begin{array} { l l } - x ^ { 2 } - 2 a x - a , & x < 0 , \\ \mathrm { e } ^ { x } + \ln ( x + 1 ) , & x \geqslant 0 \end{array} \right.$ is monotonically increasing on $\mathbb { R }$ , then the range of $a$ is
A. $( - \infty , 0 ]$
B. $[ - 1,0 ]$
C. $[ - 1,1 ]$
D. $[ 0 , + \infty )$

Let function $f ( x ) = ( x - 1 ) ^ { 2 } ( x - 4 )$ , then
A. $x = 3$ is a local minimum point of $f ( x )$
B. When $0 < x < 1$ , $f ( x ) < f \left( x ^ { 2 } \right)$
C. When $1 < x < 2$ , $- 4 < f ( 2 x - 1 ) < 0$
D. When $- 1 < x < 0$ , $f ( 2 - x ) > f ( x )$

Let $f ( x ) = 2 x ^ { 3 } - 3 a x ^ { 2 } + 1$. Then
A. When $a > 1$, $f ( x )$ has three zeros
B. When $a < 0$, $x = 0$ is a local maximum point of $f ( x )$
C. There exist $a , b$ such that $x = b$ is an axis of symmetry of the curve $y = f ( x )$
D. There exists $a$ such that the point $( 1 , f ( 1 ) )$ is a center of symmetry of the curve $y = f ( x )$

If $x = 2$ is an extremum point of the function $f(x) = (x-1)(x-2)(x-a)$, then $f(0) = $ \_\_\_\_

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

We assume that $f$ is the zero application on $C(0,1)$ and that $u$ is an element of $\mathcal{D}_f$. For all $n \in \mathbb{N}$, we define the application $$u_n : \begin{array}{rll} \bar{D}(0,1) & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & u(x,y) + \dfrac{1}{n}(x^2 + y^2) \end{array}$$
Suppose that $u_n$ admits a local maximum at $(\tilde{x}, \tilde{y}) \in D(0,1)$.
a) By examining the behavior of the function $x \mapsto u_n(x, \tilde{y})$ show that, in this case, $\partial_{11} u_n(\tilde{x}, \tilde{y}) \leqslant 0$. Similarly, one can show that $\partial_{22} u_n(\tilde{x}, \tilde{y}) \leqslant 0$. Thus $\Delta u_n(\tilde{x}, \tilde{y}) \leqslant 0$. This result is admitted for the rest.
b) Deduce that $u_n$ does not admit a local maximum on $D(0,1)$.

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$) and $f: U \to \mathbb{R}$ of class $\mathcal{C}^2$. Let $f$ be a function continuous on $\bar{U}$. Show that $f$ attains a maximum at some point $x_0 \in \bar{U}$.