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. Given the function $f ( x ) = \frac { a x } { ( x + r ) ^ { 2 } } ( a > 0 , r > 0 )$
(1) Find the domain of $f ( x )$ and discuss the monotonicity of $f ( x )$;
(2) If $\frac { a } { r } = 400$, find the extreme values of $f ( x )$ on $( 0 , + \infty )$.

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 $x$

Given the function $f(x) = \ln x + \ln(2 - x)$, then
A. $f(x)$ is monotonically increasing on $(0, 2)$
B. $f(x)$ is monotonically decreasing on $(0, 2)$
C. $f(x)$ is increasing on $(0, 1)$ and decreasing on $(1, 2)$
D. $f(x)$ is decreasing on $(0, 1)$ and increasing on $(1, 2)$

(12 points)
Given the function $f(x) = e^x(e^x - a) - a^2x$.
(1) Discuss the monotonicity of $f(x)$;
(2) If $f(x) \geq 0$, find the range of values for $a$.

(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 $a$.

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 $a$.

21. Solution: (1) When $a = 0$, it clearly does not satisfy the problem conditions, so $a \neq 0$. & 1 mark \hline $f'(x) = 3ax^2 - 2x$. Setting $f'(x) = 0$, we get $x = 0$ or $x = \frac{2}{3a}$. & $f'(x) = 3ax^2 - 2x$. Setting $f'(x) = 0$, we get $x = 0$ or $x = \frac{2}{3a}$. \hline From the problem conditions, $1 < \frac{2}{3a} < 3$. Solving, we get $\frac{2}{9} < a < \frac{2}{3}$, i.e., the range of $a$ is $\left(\frac{2}{9}, \frac{2}{3}\right)$. & 4 marks \hline (2) When $a = 0$, $f(x) = -x^2$ has minimum value $f(2) = -4$ on $[-1, 2]$. & 5 marks \hline Since $x \in [-1, 2]$, $\therefore 3x - \frac{2}{a} \leq 0$. & When $0 < a \leq \frac{1}{3}$, $\frac{2}{a} \geq 6$. $f'(x) = ax\left(3x - \frac{2}{a}\right)$. \hline 6 marks & \hline $\because f(2) - f(-1) = (8a - 4) - (-a - 1) = 9a - 3 \leq 0$, $\therefore f(x)_{\min} = f(2) = 8a - 4$. & 7 marks \hline When $a > \frac{1}{3}$, $f'(x) = ax\left(3x - \frac{2}{a}\right)$, $0 < \frac{2}{3a} < 2$. When $x \in [-1, 0) \cup \left(\frac{2}{3a}, 2\right]$, $f'(x) > 0$; & \hline When $x \in \left(0, \frac{2}{3a}\right)$, $f'(x) < 0$. & 8 marks \hline $\therefore f(x)_{\min} = \min\left\{f(-1), f\left(\frac{2}{3a}\right)\right\}$. & \hline 9 marks & \hline Since $a > \frac{1}{3}$, $\therefore 27a^3 + 27a^2 - 4 > 0$, $\frac{27a^3 + 27a^2 - 4}{27a^2} > 0$. & 10 marks \hline $\therefore f(x)_{\min} = f(-1) = -a - 1$. & 11 marks \hline In summary, when $0 \leq a \leq \frac{1}{3}$, $f(x)_{\min} = 8a - 4$; when $a > \frac{1}{3}$, $f(x)_{\min} = -a - 1$. & 12 marks \hline 1 mark & \hline

Given the function $f ( x ) = \mathrm { e } ^ { x } - a ( x + 2 )$ .
(1) When $a = 1$ , discuss the monotonicity of $f ( x )$ ;
(2) If $f ( x )$ has two zeros, find the range of values for $a$ .

Given the function $f ( x ) = x ^ { 3 } - k x + k ^ { 2 }$ .
(1) Discuss the monotonicity of $f ( x )$;
(2) If $f ( x )$ has three zeros, find the range of values for $k$ .

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 $y = f ( x )$

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 $\_\_\_\_$.

Given the function $f ( x ) = a x - \frac { 1 } { x } - ( a + 1 ) \ln x$ .
(1) When $a = 0$ , find the maximum value of $f ( x )$ ;
(2) If $f ( x )$ has exactly one zero point, find the range of values for $a$ .

Given the function $f ( x ) = \frac { e ^ { x } } { x } - \ln x + x - a$.
(1) If $f ( x ) \geq 0$, find the range of values for $a$;
(2) Prove that if $f ( x )$ has two zeros $x _ { 1 }$ and $x _ { 2 }$, then $x _ { 1 } x _ { 2 } < 1$.

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 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 $f ( x ) = ( x + a ) \ln ( x + b )$. 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

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 $y = f ( x )$