The graph of $f ^ { \prime }$, the derivative of the function $f$, is shown above. Which of the following statements must be true?
I. $f$ has a relative minimum at $x = - 3$.
II. The graph of $f$ has a point of inflection at $x = - 2$.
III. The graph of $f$ is concave down for $0 < x < 4$.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

The figure above shows the graph of $f ^ { \prime }$, the derivative of a twice-differentiable function $f$, on the closed interval $0 \leq x \leq 8$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = 1$, $x = 3$, and $x = 5$. The areas of the regions between the graph of $f ^ { \prime }$ and the $x$-axis are labeled in the figure. The function $f$ is defined for all real numbers and satisfies $f ( 8 ) = 4$.
(a) Find all values of $x$ on the open interval $0 < x < 8$ for which the function $f$ has a local minimum. Justify your answer.
(b) Determine the absolute minimum value of $f$ on the closed interval $0 \leq x \leq 8$. Justify your answer.
(c) On what open intervals contained in $0 < x < 8$ is the graph of $f$ both concave down and increasing? Explain your reasoning.
(d) The function $g$ is defined by $g ( x ) = ( f ( x ) ) ^ { 3 }$. If $f ( 3 ) = - \frac { 5 } { 2 }$, find the slope of the line tangent to the graph of $g$ at $x = 3$.

The function $f$ is defined on the closed interval $[-2, 8]$ and satisfies $f(2) = 1$. The graph of $f'$, the derivative of $f$, consists of two line segments and a semicircle, as shown in the figure.
(a) Does $f$ have a relative minimum, a relative maximum, or neither at $x = 6$? Give a reason for your answer.
(b) On what open intervals, if any, is the graph of $f$ concave down? Give a reason for your answer.
(c) Find the value of $\lim_{x \rightarrow 2} \frac{6f(x) - 3x}{x^{2} - 5x + 6}$, or show that it does not exist. Justify your answer.
(d) Find the absolute minimum value of $f$ on the closed interval $[-2, 8]$. Justify your answer.

Below is the graphical representation of $f^{\prime}$, the derivative function of a function $f$ defined on [0;7].
The variation table of $f$ on the interval [0; 7] is:
a.

$x$	0	3,25	7
$f(x)$

b.

$x$	0	2	5	7
$f(x)$

c.

$x$	0	2	5	7
$f(x)$	$\nearrow$

d.

$x$	0	2	7
$f(x)$

The function $y = f ( x )$ is continuous on all real numbers, and for all $x$ with $| x | \neq 1$, the derivative $f ^ { \prime } ( x )$ is
$$f ^ { \prime } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & ( | x | < 1 ) \\ - 1 & ( | x | > 1 ) \end{array} \right.$$
Which of the following statements in are true? [3 points]

ㄱ. The function $y = f ( x )$ has an extremum at $x = - 1$. ㄴ. For all real numbers $x$, $f ( x ) = f ( - x )$. ㄷ. If $f ( 0 ) = 0$, then $f ( 1 ) > 0$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

A cubic function $f ( x )$ satisfies $f ( 0 ) > 0$. Define the function $g ( x )$ as
$$g ( x ) = \left| \int _ { 0 } ^ { x } f ( t ) d t \right|$$
The graph of the function $y = g ( x )$ is as shown in the figure. Which of the following statements are correct? Choose all that apply. [4 points]

ㄱ. The equation $f ( x ) = 0$ has three distinct real roots. ㄴ. $f ^ { \prime } ( 0 ) < 0$ ㄷ. The number of natural numbers $m$ satisfying $\int _ { m } ^ { m + 2 } f ( x ) d x > 0$ is 3.
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

A real coefficient polynomial $f(x)$ has degree greater than 5, and its leading coefficient is positive. Moreover, $f(x)$ has local minima at $x = 1, 2, 4$ and local maxima at $x = 3, 5$. Based on the above, select the correct options.
(1) $f(1) < f(3)$
(2) There exist real numbers $a, b$ satisfying $1 < a < b < 2$ such that $f'(a) > 0$ and $f'(b) < 0$
(3) $f''(3) > 0$
(4) There exists a real number $c > 5$ such that $f'(c) > 0$
(5) The degree of $f(x)$ is greater than 7

The function $\mathrm { f } ( x )$ has derivative $\mathrm { f } ^ { \prime } ( x )$.
The diagram below shows the graph of $y = f ^ { \prime } ( x )$.
Which point corresponds to a local minimum of $\mathrm { f } ( x )$ ?
[Figure]

The graph of the derivative function $f ^ { \prime }$ of a function f defined on the set of real numbers is given below.
Accordingly, regarding the function f: I. It is decreasing. II. $f ( a )$ is a local maximum value. III. $f ^ { \prime \prime } ( a )$ is not defined.
Which of the following statements are true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

Let $f$ be a function defined on the set of real numbers, and let the derivative of $f$ be denoted by $f ^ { \prime }$. The graph of the function $f ^ { \prime }$ is the parabolic curve shown in the figure.
Accordingly, regarding the function f: I. $f ( 0 ) < 0$ II. It is decreasing on the interval (-a, a). III. $f ( a )$ is a local minimum value.
Which of the following statements are definitely true?
A) Only II
B) Only III
C) I and II
D) II and III
E) I, II and III

The graph of the derivative function $f ^ { \prime }$ of a function f defined on the set of real numbers is given in the following Cartesian coordinate plane.
Accordingly; what is the correct ordering of the values $\mathbf { f } ( \mathbf { 0 } )$, $\mathbf { f } ( \mathbf { 1 } )$ and $\mathbf { f } ( \mathbf { 2 } )$?
A) $\mathrm { f } ( 0 ) < \mathrm { f } ( 1 ) < \mathrm { f } ( 2 )$ B) $\mathrm { f } ( 0 ) < \mathrm { f } ( 2 ) < \mathrm { f } ( 1 )$ C) $f ( 1 ) < f ( 2 ) < f ( 0 )$ D) $\mathrm { f } ( 2 ) < \mathrm { f } ( 0 ) < \mathrm { f } ( 1 )$ E) $\mathrm { f } ( 2 ) < \mathrm { f } ( 1 ) < \mathrm { f } ( 0 )$

In the rectangular coordinate plane, the graph of the derivative $f'$ of a continuous function $f$ defined on the set of real numbers is shown in the figure.
$$f(5) = f(20) = 0$$
Given that, what is the local minimum value of the function $f$?
A) $-18$ B) $-15$ C) $-12$ D) $-9$ E) $-6$