The graph of $f ^ { \prime }$, the derivative of the function $f$, is shown above. Which of the following could be the graph of $f$?
(A), (B), (C), (D) [graphs as shown in the exam]

$x$	- 1	0	2	4	5
$f ^ { \prime } ( x )$	11	9	8	5	2

Let $f$ be a twice-differentiable function. Values of $f ^ { \prime }$, the derivative of $f$, at selected values of $x$ are given in the table above. Which of the following statements must be true?
(A) $f$ is increasing for $- 1 \leq x \leq 5$.
(B) The graph of $f$ is concave down for $- 1 < x < 5$.
(C) There exists $c$, where $- 1 < c < 5$, such that $f ^ { \prime } ( c ) = - \frac { 3 } { 2 }$.
(D) There exists $c$, where $- 1 < c < 5$, such that $f ^ { \prime \prime } ( c ) = - \frac { 3 } { 2 }$.

The figure above shows the graph of $f ^ { \prime }$, the derivative of a twice-differentiable function $f$, on the closed interval $[ 0,4 ]$. The areas of the regions bounded by the graph of $f ^ { \prime }$ and the $x$-axis on the intervals $[ 0,1 ] , [ 1,2 ] , [ 2,3 ]$, and $[ 3,4 ]$ are $2, 6, 10$, and $14$, respectively. The graph of $f ^ { \prime }$ has horizontal tangents at $x = 0.6 , x = 1.6$, $x = 2.5$, and $x = 3.5$. It is known that $f ( 2 ) = 5$.
(a) On what open intervals contained in $( 0,4 )$ is the graph of $f$ both decreasing and concave down? Give a reason for your answer.
(b) Find the absolute minimum value of $f$ on the interval $[ 0,4 ]$. Justify your answer.
(c) Evaluate $\int _ { 0 } ^ { 4 } f ( x ) f ^ { \prime } ( x ) \, d x$.
(d) The function $g$ is defined by $g ( x ) = x ^ { 3 } f ( x )$. Find $g ^ { \prime } ( 2 )$. Show the work that leads to your answer.

17. The graph of a twice-differentiable function $f$ is shown in the figure above. Which of the following is true?
(A) $f ( 1 ) < f ^ { \prime } ( 1 ) < f ^ { \prime \prime } ( 1 )$
(B) $f ( 1 ) < f ^ { \prime \prime } ( 1 ) < f ^ { \prime } ( 1 )$
(C) $f ^ { \prime } ( 1 ) < f ( 1 ) < f ^ { \prime \prime } ( 1 )$
(D) $f ^ { \prime \prime } ( 1 ) < f ( 1 ) < f ^ { \prime } ( 1 )$
(E) $f ^ { \prime \prime } ( 1 ) < f ^ { \prime } ( 1 ) < f ( 1 )$

The figure above shows the graph of $f ^ { \prime }$, the derivative of the function $f$, for $- 7 \leq x \leq 7$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = - 3 , x = 2$, and $x = 5$, and a vertical tangent line at $x = 3$.
(a) Find all values of $x$, for $- 7 < x < 7$, at which $f$ attains a relative minimum. Justify your answer.
(b) Find all values of $x$, for $- 7 < x < 7$, at which $f$ attains a relative maximum. Justify your answer.
(c) Find all values of $x$, for $- 7 < x < 7$, at which $f ^ { \prime \prime } ( x ) < 0$.
(d) At what value of $x$, for $- 7 \leq x \leq 7$, does $f$ attain its absolute maximum? Justify your answer.

Let $f$ be a function defined on the closed interval $-3 \leq x \leq 4$ with $f(0) = 3$. The graph of $f'$, the derivative of $f$, consists of one line segment and a semicircle, as shown above.
(a) On what intervals, if any, is $f$ increasing? Justify your answer.
(b) Find the $x$-coordinate of each point of inflection of the graph of $f$ on the open interval $-3 < x < 4$. Justify your answer.
(c) Find an equation for the line tangent to the graph of $f$ at the point $(0, 3)$.
(d) Find $f(-3)$ and $f(4)$. Show the work that leads to your answers.

The figure above shows the graph of $f'$, the derivative of the function $f$, on the closed interval $-1 \leq x \leq 5$. The graph of $f'$ has horizontal tangent lines at $x = 1$ and $x = 3$. The function $f$ is twice differentiable with $f(2) = 6$.
(a) Find the $x$-coordinate of each of the points of inflection of the graph of $f$. Give a reason for your answer.
(b) At what value of $x$ does $f$ attain its absolute minimum value on the closed interval $-1 \leq x \leq 5$? At what value of $x$ does $f$ attain its absolute maximum value on the closed interval $-1 \leq x \leq 5$? Show the analysis that leads to your answers.
(c) Let $g$ be the function defined by $g(x) = x f(x)$. Find an equation for the line tangent to the graph of $g$ at $x = 2$.

Let $f$ be a function that is continuous on the interval $[ 0,4 )$. The function $f$ is twice differentiable except at $x = 2$. The function $f$ and its derivatives have the properties indicated in the table below, where DNE indicates that the derivatives of $f$ do not exist at $x = 2$.

$x$	0	$0 < x < 1$	1	$1 < x < 2$	2	$2 < x < 3$	3	$3 < x < 4$
$f ( x )$	- 1	Negative	0	Positive	2	Positive	0	Negative
$f ^ { \prime } ( x )$	4	Positive	0	Positive	DNE	Negative	- 3	Negative
$f ^ { \prime \prime } ( x )$	- 2	Negative	0	Positive	DNE	Negative	0	Positive

(a) For $0 < x < 4$, find all values of $x$ at which $f$ has a relative extremum. Determine whether $f$ has a relative maximum or a relative minimum at each of these values. Justify your answer.
(b) On the axes provided, sketch the graph of a function that has all the characteristics of $f$.
(c) Let $g$ be the function defined by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) \, dt$ on the open interval $( 0,4 )$. For $0 < x < 4$, find all values of $x$ at which $g$ has a relative extremum. Determine whether $g$ has a relative maximum or a relative minimum at each of these values. Justify your answer.
(d) For the function $g$ defined in part (c), find all values of $x$, for $0 < x < 4$, at which the graph of $g$ has a point of inflection. Justify your answer.

Let $f$ be a function defined on the closed interval $- 5 \leq x \leq 5$ with $f ( 1 ) = 3$. The graph of $f ^ { \prime }$, the derivative of $f$, consists of two semicircles and two line segments, as shown above. (a) For $- 5 < x < 5$, find all values $x$ at which $f$ has a relative maximum. Justify your answer. (b) For $- 5 < x < 5$, find all values $x$ at which the graph of $f$ has a point of inflection. Justify your answer. (c) Find all intervals on which the graph of $f$ is concave up and also has positive slope. Explain your reasoning. (d) Find the absolute minimum value of $f ( x )$ over the closed interval $- 5 \leq x \leq 5$. Explain your reasoning.

Let $g$ be a continuous function with $g ( 2 ) = 5$. The graph of the piecewise-linear function $g ^ { \prime }$, the derivative of $g$, is shown above for $- 3 \leq x \leq 7$. (a) Find the $x$-coordinate of all points of inflection of the graph of $y = g ( x )$ for $- 3 < x < 7$. Justify your answer. (b) Find the absolute maximum value of $g$ on the interval $- 3 \leq x \leq 7$. Justify your answer. (c) Find the average rate of change of $g ( x )$ on the interval $- 3 \leq x \leq 7$. (d) Find the average rate of change of $g ^ { \prime } ( x )$ on the interval $- 3 \leq x \leq 7$. Does the Mean Value Theorem applied on the interval $- 3 \leq x \leq 7$ guarantee a value of $c$, for $- 3 < c < 7$, such that $g ^ { \prime \prime } ( c )$ is equal to this average rate of change? Why or why not?

The derivative of a function $f$ is defined by $$f'(x) = \begin{cases} g(x) & \text{for } -4 \leq x \leq 0 \\ 5e^{-x/3} - 3 & \text{for } 0 < x \leq 4 \end{cases}.$$ The graph of the continuous function $f'$, shown in the figure above, has $x$-intercepts at $x = -2$ and $x = 3\ln\left(\frac{5}{3}\right)$. The graph of $g$ on $-4 \leq x \leq 0$ is a semicircle, and $f(0) = 5$.
(a) For $-4 < x < 4$, find all values of $x$ at which the graph of $f$ has a point of inflection. Justify your answer.
(b) Find $f(-4)$ and $f(4)$.
(c) For $-4 \leq x \leq 4$, find the value of $x$ at which $f$ has an absolute maximum. Justify your answer.

Let $f$ be a function that is continuous on the interval $[ 0,4 )$. The function $f$ is twice differentiable except at $x = 2$. The function $f$ and its derivatives have the properties indicated in the table above, where DNE indicates that the derivatives of $f$ do not exist at $x = 2$.
(a) For $0 < x < 4$, find all values of $x$ at which $f$ has a relative extremum. Determine whether $f$ has a relative maximum or a relative minimum at each of these values. Justify your answer.
(b) On the axes provided, sketch the graph of a function that has all the characteristics of $f$.
(Note: Use the axes provided in the pink test booklet.)
(c) Let $g$ be the function defined by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t$ on the open interval $( 0,4 )$. For $0 < x < 4$, find all values of $x$ at which $g$ has a relative extremum. Determine whether $g$ has a relative maximum or a relative minimum at each of these values. Justify your answer.
(d) For the function $g$ defined in part (c), find all values of $x$, for $0 < x < 4$, at which the graph of $g$ has a point of inflection. Justify your answer.

The function $g$ is defined and differentiable on the closed interval $[-7, 5]$ and satisfies $g(0) = 5$. The graph of $y = g'(x)$, the derivative of $g$, consists of a semicircle and three line segments, as shown in the figure above.
(a) Find $g(3)$ and $g(-2)$.
(b) Find the $x$-coordinate of each point of inflection of the graph of $y = g(x)$ on the interval $-7 < x < 5$. Explain your reasoning.
(c) The function $h$ is defined by $h(x) = g(x) - \frac{1}{2}x^2$. Find the $x$-coordinate of each critical point of $h$, where $-7 < x < 5$, and classify each critical point as the location of a relative minimum, relative maximum, or neither a minimum nor a maximum. Explain your reasoning.

The graph of a differentiable function $f$ is shown above. If $h ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$, which of the following is true?
(A) $h ( 6 ) < h ^ { \prime } ( 6 ) < h ^ { \prime \prime } ( 6 )$
(B) $h ( 6 ) < h ^ { \prime \prime } ( 6 ) < h ^ { \prime } ( 6 )$
(C) $h ^ { \prime } ( 6 ) < h ( 6 ) < h ^ { \prime \prime } ( 6 )$
(D) $h ^ { \prime \prime } ( 6 ) < h ( 6 ) < h ^ { \prime } ( 6 )$
(E) $h ^ { \prime \prime } ( 6 ) < h ^ { \prime } ( 6 ) < h ( 6 )$

The graph of the function $f$ is shown in the figure above. For which of the following values of $x$ is $f ^ { \prime } ( x )$ positive and increasing?
(A) $a$
(B) $b$
(C) $c$
(D) $d$
(E) $e$

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 graph of $f ^ { \prime }$, the derivative of $f$, is shown in the figure above. The function $f$ has a local maximum at $x =$
(A) $-3$
(B) $-1$
(C) 1
(D) 3
(E) 4

Let $f$ be a polynomial function with values of $f ^ { \prime } ( x )$ at selected values of $x$ given in the table above.

$x$	$-2$	0	3	5	6
$f ^ { \prime } ( x )$	3	1	4	7	5

Which of the following must be true for $-2 < x < 6$ ?
(A) The graph of $f$ is concave up.
(B) The graph of $f$ has at least two points of inflection.
(C) $f$ is increasing.
(D) $f$ has no critical points.
(E) $f$ has at least two relative extrema.

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 twice-differentiable functions $f$ and $g$ are defined for all real numbers $x$. Values of $f$, $f ^ { \prime }$, $g$, and $g ^ { \prime }$ for various values of $x$ are given in the table below.

$x$	-2	$- 2 < x < - 1$	-1	$- 1 < x < 1$	1	$1 < x < 3$	3
$f ( x )$	12	Positive	8	Positive	2	Positive	7
$f ^ { \prime } ( x )$	-5	Negative	0	Negative	0	Positive	$\frac { 1 } { 2 }$
$g ( x )$	-1	Negative	0	Positive	3	Positive	1
$g ^ { \prime } ( x )$	2	Positive	$\frac { 3 } { 2 }$	Positive	0	Negative	-2

(a) Find the $x$-coordinate of each relative minimum of $f$ on the interval $[ - 2, 3 ]$. Justify your answers.
(b) Explain why there must be a value $c$, for $- 1 < c < 1$, such that $f ^ { \prime \prime } ( c ) = 0$.
(c) The function $h$ is defined by $h ( x ) = \ln ( f ( x ) )$. Find $h ^ { \prime } ( 3 )$. Show the computations that lead to your answer.
(d) Evaluate $\displaystyle\int _ { - 2 } ^ { 3 } f ^ { \prime } ( g ( x ) ) g ^ { \prime } ( x ) \, dx$.

Let $f$ be a differentiable function with $f(4) = 3$. On the interval $0 \leq x \leq 7$, the graph of $f'$, the derivative of $f$, consists of a semicircle and two line segments, as shown in the figure above.
(a) Find $f(0)$ and $f(5)$.
(b) Find the $x$-coordinates of all points of inflection of the graph of $f$ for $0 < x < 7$. Justify your answer.
(c) Let $g$ be the function defined by $g(x) = f(x) - x$. On what intervals, if any, is $g$ decreasing for $0 \leq x \leq 7$? Show the analysis that leads to your answer.
(d) For the function $g$ defined in part (c), find the absolute minimum value on the interval $0 \leq x \leq 7$. Justify your answer.

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 $\lim_{x \to 2} \frac{6f(x) - 3x}{x^{2} - 5x + 6}$, 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.

4. Let $f$ be a function defined on the closed interval $- 3 \leq x \leq 4$ with $f ( 0 ) = 3$. The graph of $f ^ { \prime }$, the derivative of $f$, consists of one line segment and a semicircle, as shown above.
(a) On what intervals, if any, is $f$ increasing? Justify your answer.
(b) Find the $x$-coordinate of each point of inflection of the graph of $f$ on the open interval $- 3 < x < 4$. Justify your answer.
(c) Find an equation for the line tangent to the graph of $f$ at the point ( 0,3 ).
(d) Find $f ( - 3 )$ and $f ( 4 )$. Show the work that leads to your answers. [Figure]

The figure above shows the graph of $f ^ { \prime }$, the derivative of the function $f$, on the closed interval $- 1 \leq x \leq 5$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = 1$ and $x = 3$. The function $f$ is twice differentiable with $f ( 2 ) = 6$.
(a) Find the $x$-coordinate of each of the points of inflection of the graph of $f$. Give a reason for your answer.
(b) At what value of $x$ does $f$ attain its absolute minimum value on the closed interval $- 1 \leq x \leq 5$ ? At what value of $x$ does $f$ attain its absolute maximum value on the closed interval $- 1 \leq x \leq 5$ ? Show the analysis that leads to your answers.
(c) Let $g$ be the function defined by $g ( x ) = x f ( x )$. Find an equation for the line tangent to the graph of $g$ at $x = 2$.

Let $f$ be a function defined on the closed interval $- 5 \leq x \leq 5$ with $f ( 1 ) = 3$. The graph of $f ^ { \prime }$, the derivative of $f$, consists of two semicircles and two line segments, as shown above. (a) For $- 5 < x < 5$, find all values $x$ at which $f$ has a relative maximum. Justify your answer. (b) For $- 5 < x < 5$, find all values $x$ at which the graph of $f$ has a point of inflection. Justify your answer. (c) Find all intervals on which the graph of $f$ is concave up and also has positive slope. Explain your reasoning. (d) Find the absolute minimum value of $f ( x )$ over the closed interval $- 5 \leq x \leq 5$. Explain your reasoning.