Question 1

Find the x-value at which the given function has a point of inflection. $y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 15 x + 9$

Accepted Answer

A)  $x = 0$ 
B)  $x = 5$ 
C)  $x = 4$ 
D)  $x = 3$ 
E) no point of inflection 
A)  $x = 0$ 
B)  $x = 5$ 
C)  $x = 4$ 
D)  $x = 3$ 
E) no point of inflection

Question 2

Identify the open intervals where the function $f ( x ) = 4 x ^ { 2 } + 6 x - 1$ is increasing or decreasing.

Accepted Answer

A) decreasing: $\left( - \infty , - \frac { 3 } { 4 } \right)$ ; increasing: $\left( - \frac { 3 } { 4 } , \infty \right)$ 
B) increasing: $\left( - \infty , - \frac { 3 } { 4 } \right)$ ; decreasing: $\left( - \frac { 3 } { 4 } , \infty \right)$ 
C) increasing on $( - \infty , \infty )$ 
D) decreasing on $( - \infty , \infty )$ 
E) none of the above 
A) decreasing: $\left( - \infty , - \frac { 3 } { 4 } \right)$ ; increasing: $\left( - \frac { 3 } { 4 } , \infty \right)$ 
B) increasing: $\left( - \infty , - \frac { 3 } { 4 } \right)$ ; decreasing: $\left( - \frac { 3 } { 4 } , \infty \right)$ 
C) increasing on $( - \infty , \infty )$ 
D) decreasing on $( - \infty , \infty )$ 
E) none of the above

Question 3

Determine the open intervals on which the graph of $f ( x ) = 2 x ^ { 2 } + 5 x - 4$ is concave downward or concave upward.

Accepted Answer

A) concave upward on $( - \infty , 0 )$ ; concave downward on $( 0 , \infty )$ 
B) concave downward on $( - \infty , \infty )$ 
C) concave upward on $( - \infty , \infty )$ 
D) concave downward on $( - \infty , 0 )$ ; concave upward on $( 0 , \infty )$ 
E) concave upward on $( - \infty , 1 )$ ; concave downward on $( 1 , \infty )$ 
A) concave upward on $( - \infty , 0 )$ ; concave downward on $( 0 , \infty )$ 
B) concave downward on $( - \infty , \infty )$ 
C) concave upward on $( - \infty , \infty )$ 
D) concave downward on $( - \infty , 0 )$ ; concave upward on $( 0 , \infty )$ 
E) concave upward on $( - \infty , 1 )$ ; concave downward on $( 1 , \infty )$

Question 4

Locate the absolute extrema of the given function on the closed interval [-48,48]. $f ( x ) = \frac { 48 x } { x ^ { 2 } + 16 }$

Accepted Answer

A) absolute max: f(4)= 6 
B) absolute min: f(-4)= -6 
C) no absolute max 
D) no absolute min 
E) both A and D
F)both A and B 
A) absolute max: f(4)= 6 
B) absolute min: f(-4)= -6 
C) no absolute max 
D) no absolute min 
E) both A and D
F)both A and B

Question 5

For the given function, find all critical numbers. $y = x ^ { 3 } - 3 x ^ { 2 } - 105 x + 5$

Accepted Answer

A)  $x = 0$ 
B)  $x = - 7$ and $x = - 5$ 
C)  $x = - 7$ and $x = 5$ 
D)  $x = - 5$ and $x = 7$ 
E)  $x = 5$ and $x = 7$ 
A)  $x = 0$ 
B)  $x = - 7$ and $x = - 5$ 
C)  $x = - 7$ and $x = 5$ 
D)  $x = - 5$ and $x = 7$ 
E)  $x = 5$ and $x = 7$

Question 6

Find all relative minima of the given function. $y = x ^ { 4 } - 8 x ^ { 3 } + 16 x ^ { 2 } + 16$

Accepted Answer

A)  $( 0,16 )$ 
B)  $( 2,32 )$ 
C)  $( 4,16 )$ 
D)  $( 0,16 )$ , $( 4,16 )$ 
E) no relative minima 
A)  $( 0,16 )$ 
B)  $( 2,32 )$ 
C)  $( 4,16 )$ 
D)  $( 0,16 )$ , $( 4,16 )$ 
E) no relative minima

Question 7

Graph a function on the interval $[ - 1,5 ]$ having the following characteristics. Absolute maximum at $x = 4.75$ Absolute minimum at $x = - 1$ Relative maximum at $x = 0.3$ Relative minimum at $x = 3.1$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 8

Locate the absolute extrema of the function $f ( x ) = x ^ { 3 } - 3 x$ on the closed interval [0,2].

Accepted Answer

A) absolute max: f(2)= 2 ; absolute min: f(1)= -2 
B) absolute max: f(1)= -2 ; absolute min: f(2)= 2 
C) absolute max: f(2)= 2 ; no absolute min 
D) no absolute max; absolute min: f(2)= 2 
E) no absolute max or min 
A) absolute max: f(2)= 2 ; absolute min: f(1)= -2 
B) absolute max: f(1)= -2 ; absolute min: f(2)= 2 
C) absolute max: f(2)= 2 ; no absolute min 
D) no absolute max; absolute min: f(2)= 2 
E) no absolute max or min

Question 9

Given $x y = 6,$ find $\frac { d y } { d t }$ when x = 4 and $\frac { d x } { d t } = - 2$

Accepted Answer

A)  $\frac { d y } { d t } = \frac { 27 } { 4 }$ 
B)  $\frac { d y } { d t } = - \frac { 3 } { 4 }$ 
C)  $\frac { d y } { d t } = \frac { 3 } { 4 }$ 
D)  $\frac { d y } { d t } = \frac { 4 } { 3 }$ 
E)  $\frac { d y } { d t } = - \frac { 4 } { 27 }$ 
A)  $\frac { d y } { d t } = \frac { 27 } { 4 }$ 
B)  $\frac { d y } { d t } = - \frac { 3 } { 4 }$ 
C)  $\frac { d y } { d t } = \frac { 3 } { 4 }$ 
D)  $\frac { d y } { d t } = \frac { 4 } { 3 }$ 
E)  $\frac { d y } { d t } = - \frac { 4 } { 27 }$

Question 10

Find $y ^ { \prime }$ implicitly for $8 x ^ { 9 } - y ^ { 9 } = 5$

Accepted Answer

A)  $y ^ { \prime } = \frac { 8 x ^ { 9 } } { y ^ { 9 } }$ 
B)  $y ^ { \prime } = \frac { y ^ { 9 } } { 8 x ^ { 9 } }$ 
C)  $y ^ { \prime } = \frac { 8 x ^ { 8 } } { y ^ { 8 } }$ 
D)  $y ^ { \prime } = \frac { y ^ { 8 } } { 8 x ^ { 8 } }$ 
E)  $y ^ { \prime } = \frac { x ^ { 8 } } { 8 y ^ { 8 } }$ 
A)  $y ^ { \prime } = \frac { 8 x ^ { 9 } } { y ^ { 9 } }$ 
B)  $y ^ { \prime } = \frac { y ^ { 9 } } { 8 x ^ { 9 } }$ 
C)  $y ^ { \prime } = \frac { 8 x ^ { 8 } } { y ^ { 8 } }$ 
D)  $y ^ { \prime } = \frac { y ^ { 8 } } { 8 x ^ { 8 } }$ 
E)  $y ^ { \prime } = \frac { x ^ { 8 } } { 8 y ^ { 8 } }$

Question 11

Suppose the number y of medical degrees conferred in the United States can be modeled by $y = 0.813 t ^ { 3 } - 55.70 t ^ { 2 } + 1185.2 t + 7752,$ for $0 \leq t \leq 32$ , where t is the time in years, with $t = 0$ corresponding to 1972. Use the test for increasing and decreasing functions to estimate the years during which the number of medical degrees is increasing and the years during which it is decreasing.

Accepted Answer

A) The number of medical degrees is increasing from 1972 to 1989 and 1997 to 2002, and decreasing during 1989 to 1997. 
B) The number of medical degrees is increasing from 1972 to 1988 and 1996 to 2002, and decreasing during 1988 to 1996. 
C) The number of medical degrees is increasing from 1972 to 1989 and 1996 to 2002, and decreasing during 1989 to 1996. 
D) The number of medical degrees is increasing from 1972 to 1990 and 1996 to 2002, and decreasing during 1990 to 1996. 
E) The number of medical degrees is increasing from 1972 to 1989 and 1995 to 2002, and decreasing during 1989 to 1995. 
A) The number of medical degrees is increasing from 1972 to 1989 and 1997 to 2002, and decreasing during 1989 to 1997. 
B) The number of medical degrees is increasing from 1972 to 1988 and 1996 to 2002, and decreasing during 1988 to 1996. 
C) The number of medical degrees is increasing from 1972 to 1989 and 1996 to 2002, and decreasing during 1989 to 1996. 
D) The number of medical degrees is increasing from 1972 to 1990 and 1996 to 2002, and decreasing during 1990 to 1996. 
E) The number of medical degrees is increasing from 1972 to 1989 and 1995 to 2002, and decreasing during 1989 to 1995.

Question 12

Find the third derivative. $y = \frac { 4 } { x ^ { 9 } }$

Accepted Answer

A)  $\frac { - 3960 } { x ^ { 11 } }$ 
B)  $\frac { 3960 } { x ^ { 12 } }$ 
C)  $0$ 
D)  $\frac { 440 } { x ^ { 11 } }$ 
E)  $\frac { - 3960 } { x ^ { 12 } }$ 
A)  $\frac { - 3960 } { x ^ { 11 } }$ 
B)  $\frac { 3960 } { x ^ { 12 } }$ 
C)  $0$ 
D)  $\frac { 440 } { x ^ { 11 } }$ 
E)  $\frac { - 3960 } { x ^ { 12 } }$

Question 13

State the signs of $f ^ { \prime } ( x )$ and $f ^ { \prime \prime } ( x )$ on the interval (0, 2).

Accepted Answer

A)  $\begin{array} { l } 
f ^ { \prime } = 0 \
f ^ { \prime \prime }  0 \
f ^ { \prime \prime }  0
\end{array}$ 
E)  $\begin{array} { l } 
f ^ { \prime } > 0 \
f ^ { \prime \prime } > 0
\end{array}$ 
A)  $\begin{array} { l } 
f ^ { \prime } = 0 \
f ^ { \prime \prime }  0 \
f ^ { \prime \prime }  0
\end{array}$ 
E)  $\begin{array} { l } 
f ^ { \prime } > 0 \
f ^ { \prime \prime } > 0
\end{array}$

Question 14

Find the absolute extrema of the function $h ( t ) = ( t - 3 ) ^ { 2 / 3 }$ on the closed interval $[ - 1,6 ]$ . Round your answer to two decimal places.

Accepted Answer

A) The maximum of the function is 1 and the minimum of the function is 0. 
B) The maximum of the function is 2.52 and the minimum of the function is 1. 
C) The maximum of the function is 2.52 and the minimum of the function is 0. 
D) The maximum of the function is1 and the minimum of the function is 2.08. 
E) The maximum of the function is 0 and the minimum of the function is 2.08. 
A) The maximum of the function is 1 and the minimum of the function is 0. 
B) The maximum of the function is 2.52 and the minimum of the function is 1. 
C) The maximum of the function is 2.52 and the minimum of the function is 0. 
D) The maximum of the function is1 and the minimum of the function is 2.08. 
E) The maximum of the function is 0 and the minimum of the function is 2.08.

Question 15

Find the absolute extrema of the function $f ( x ) = \frac { 4 x } { x ^ { 2 } + 1 }$ on the interval $[ 0 , \infty )$ .

Accepted Answer

A) The maximum of the function is 1 and the minimum of the function is 0. 
B) The maximum of the function is 0 and the minimum of the function is -2. 
C) The maximum of the function is -2 and the minimum of the function is 0. 
D) The maximum of the function is 2 and the minimum of the function is 0. 
E) The maximum of the function is 0 and the minimum of the function is 2. 
A) The maximum of the function is 1 and the minimum of the function is 0. 
B) The maximum of the function is 0 and the minimum of the function is -2. 
C) The maximum of the function is -2 and the minimum of the function is 0. 
D) The maximum of the function is 2 and the minimum of the function is 0. 
E) The maximum of the function is 0 and the minimum of the function is 2.

Question 16

A function and its graph are given. Use the second derivative to locate all x-values of points of inflection on the graph of $y = f ( x )$ . Check these results against the graph shown. $$y = 18 x ^ { 4 } - 54 x ^ { 2 } + 36$$

Accepted Answer

A)  $x = - \frac { \sqrt { 2 } } { 2 }$ 
B)  $x = 0$ 
C)  $x = \frac { \sqrt { 2 } } { 2 }$ 
D)  $x = \frac { \sqrt { 2 } } { 2 }$ , $x = - \frac { \sqrt { 2 } } { 2 }$ 
E)  $x = - \frac { \sqrt { 2 } } { 2 }$ , $x = 0$ , $x = - \frac { \sqrt { 2 } } { 2 }$ 
A)  $x = - \frac { \sqrt { 2 } } { 2 }$ 
B)  $x = 0$ 
C)  $x = \frac { \sqrt { 2 } } { 2 }$ 
D)  $x = \frac { \sqrt { 2 } } { 2 }$ , $x = - \frac { \sqrt { 2 } } { 2 }$ 
E)  $x = - \frac { \sqrt { 2 } } { 2 }$ , $x = 0$ , $x = - \frac { \sqrt { 2 } } { 2 }$

Question 17

Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by $x ( t ) = \frac { 3000 t } { t ^ { 2 } + 12 }$ $t > 0$ . Find the maximum value of x. Round your answer to two decimal places.

Accepted Answer

A) 3.46 mg 
B) 433.01 mg 
C) 952.63 mg 
D) 10.74 mg 
E) 1125.30 mg 
A) 3.46 mg 
B) 433.01 mg 
C) 952.63 mg 
D) 10.74 mg 
E) 1125.30 mg

Question 18

Find any critical numbers of the function $g ( t ) = t \sqrt { 14 - t }$ , t < 14.

Accepted Answer

A) 0 
B)  $- \frac { 28 } { 3 }$ 
C)  $\frac { 28 } { 3 }$ 
D) both A and B 
E) both A and C 
A) 0 
B)  $- \frac { 28 } { 3 }$ 
C)  $\frac { 28 } { 3 }$ 
D) both A and B 
E) both A and C

Question 19

Locate the absolute extrema of the function $f ( x ) = - 4 x ^ { 2 } + 8 x - 2$ on the closed interval $[ - 2,2 ]$ .

Accepted Answer

A) no absolute max; absolute min: f(1)= 2 
B) absolute max: f(-2)= -34 ; absolute min: f(1)= 2 
C) absolute max: f(1)= 2 ; no absolute min 
D) absolute max: f(1)= 2 ; absolute min: f(-2)= -34 
E) no absolute max or min 
A) no absolute max; absolute min: f(1)= 2 
B) absolute max: f(-2)= -34 ; absolute min: f(1)= 2 
C) absolute max: f(1)= 2 ; no absolute min 
D) absolute max: f(1)= 2 ; absolute min: f(-2)= -34 
E) no absolute max or min

Question 20

Find the open intervals on which the function $y = \left\{ \begin{array} { l } 
2 x + 7 , x \leq 7 \
4 - x ^ { 2 } , x > 7
\end{array} \right.$ is increasing or decreasing.

Accepted Answer

A) The function is increasing on the interval $- \infty < x < 0$ and decreasing on the interval $0 < x < \infty$ . 
B) The function is increasing on the interval $7 < x < \infty$ and decreasing on the interval $- \infty < x < 7$ . 
C) The function is increasing on the interval $- \infty < x \leq 7$ and decreasing on the interval $7 < x < \infty$ . 
D) The function is increasing on the interval $0 < x < \infty$ and decreasing on the interval $- \infty < x < 0$ . 
E) The function is increasing on the interval $- \infty < x < 7$ and decreasing on the interval $7 < x < \infty$ . 
A) The function is increasing on the interval $- \infty < x < 0$ and decreasing on the interval $0 < x < \infty$ . 
B) The function is increasing on the interval $7 < x < \infty$ and decreasing on the interval $- \infty < x < 7$ . 
C) The function is increasing on the interval $- \infty < x \leq 7$ and decreasing on the interval $7 < x < \infty$ . 
D) The function is increasing on the interval $0 < x < \infty$ and decreasing on the interval $- \infty < x < 0$ . 
E) The function is increasing on the interval $- \infty < x < 7$ and decreasing on the interval $7 < x < \infty$ .

Find the x-value at which the given function has a point of inflection. $y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 15 x + 9$

Identify the open intervals where the function $f ( x ) = 4 x ^ { 2 } + 6 x - 1$ is increasing or decreasing.

Determine the open intervals on which the graph of $f ( x ) = 2 x ^ { 2 } + 5 x - 4$ is concave downward or concave upward.

Locate the absolute extrema of the given function on the closed interval [-48,48]. $f ( x ) = \frac { 48 x } { x ^ { 2 } + 16 }$

For the given function, find all critical numbers. $y = x ^ { 3 } - 3 x ^ { 2 } - 105 x + 5$

Find all relative minima of the given function. $y = x ^ { 4 } - 8 x ^ { 3 } + 16 x ^ { 2 } + 16$

Graph a function on the interval $[ - 1,5 ]$ having the following characteristics. Absolute maximum at $x = 4.75$ Absolute minimum at $x = - 1$ Relative maximum at $x = 0.3$ Relative minimum at $x = 3.1$

Locate the absolute extrema of the function $f ( x ) = x ^ { 3 } - 3 x$ on the closed interval [0,2].

Given $x y = 6,$ find $\frac { d y } { d t }$ when x = 4 and $\frac { d x } { d t } = - 2$

Find $y ^ { \prime }$ implicitly for $8 x ^ { 9 } - y ^ { 9 } = 5$

Find the third derivative. $y = \frac { 4 } { x ^ { 9 } }$

State the signs of $f ^ { \prime } ( x )$ and $f ^ { \prime \prime } ( x )$ on the interval (0, 2). $State the signs of f ^ { \prime } ( x ) and f ^ { \prime \prime } ( x ) on the interval (0, 2).$

Find the absolute extrema of the function $h ( t ) = ( t - 3 ) ^ { 2 / 3 }$ on the closed interval $[ - 1,6 ]$ . Round your answer to two decimal places.

Find the absolute extrema of the function $f ( x ) = \frac { 4 x } { x ^ { 2 } + 1 }$ on the interval $[ 0 , \infty )$ .

Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by $x ( t ) = \frac { 3000 t } { t ^ { 2 } + 12 }$ $t > 0$ . Find the maximum value of x. Round your answer to two decimal places.

Find any critical numbers of the function $g ( t ) = t \sqrt { 14 - t }$ , t < 14.

Locate the absolute extrema of the function $f ( x ) = - 4 x ^ { 2 } + 8 x - 2$ on the closed interval $[ - 2,2 ]$ .

Find the open intervals on which the function $y = \left\{ \begin{array} { l } 2 x + 7 , x \leq 7 \\4 - x ^ { 2 } , x > 7\end{array} \right.$ is increasing or decreasing.

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Exam 9: Applications of the Derivative

Find the x-value at which the given function has a point of inflection. y=13x3−4x2+15x+9y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 15 x + 9y=31​x3−4x2+15x+9

Identify the open intervals where the function f(x)=4x2+6x−1f ( x ) = 4 x ^ { 2 } + 6 x - 1f(x)=4x2+6x−1 is increasing or decreasing.

Determine the open intervals on which the graph of f(x)=2x2+5x−4f ( x ) = 2 x ^ { 2 } + 5 x - 4f(x)=2x2+5x−4 is concave downward or concave upward.

Locate the absolute extrema of the given function on the closed interval [-48,48]. f(x)=48xx2+16f ( x ) = \frac { 48 x } { x ^ { 2 } + 16 }f(x)=x2+1648x​

For the given function, find all critical numbers. y=x3−3x2−105x+5y = x ^ { 3 } - 3 x ^ { 2 } - 105 x + 5y=x3−3x2−105x+5

Find all relative minima of the given function. y=x4−8x3+16x2+16y = x ^ { 4 } - 8 x ^ { 3 } + 16 x ^ { 2 } + 16y=x4−8x3+16x2+16

Graph a function on the interval [−1,5][ - 1,5 ][−1,5] having the following characteristics. Absolute maximum at x=4.75x = 4.75x=4.75 Absolute minimum at x=−1x = - 1x=−1 Relative maximum at x=0.3x = 0.3x=0.3 Relative minimum at x=3.1x = 3.1x=3.1

Locate the absolute extrema of the function f(x)=x3−3xf ( x ) = x ^ { 3 } - 3 xf(x)=x3−3x on the closed interval [0,2].

Given xy=6,x y = 6,xy=6, find dydt\frac { d y } { d t }dtdy​ when x = 4 and dxdt=−2\frac { d x } { d t } = - 2dtdx​=−2

Find y′y ^ { \prime }y′ implicitly for 8x9−y9=58 x ^ { 9 } - y ^ { 9 } = 58x9−y9=5

Find the third derivative. y=4x9y = \frac { 4 } { x ^ { 9 } }y=x94​

State the signs of f′(x)f ^ { \prime } ( x )f′(x) and f′′(x)f ^ { \prime \prime } ( x )f′′(x) on the interval (0, 2).

Find the absolute extrema of the function h(t)=(t−3)2/3h ( t ) = ( t - 3 ) ^ { 2 / 3 }h(t)=(t−3)2/3 on the closed interval [−1,6][ - 1,6 ][−1,6] . Round your answer to two decimal places.

Find the absolute extrema of the function f(x)=4xx2+1f ( x ) = \frac { 4 x } { x ^ { 2 } + 1 }f(x)=x2+14x​ on the interval [0,∞)[ 0 , \infty )[0,∞) .

A function and its graph are given. Use the second derivative to locate all x-values of points of inflection on the graph of y=f(x)y = f ( x )y=f(x) . Check these results against the graph shown. y=18x4−54x2+36y = 18 x ^ { 4 } - 54 x ^ { 2 } + 36y=18x4−54x2+36

Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by x(t)=3000tt2+12x ( t ) = \frac { 3000 t } { t ^ { 2 } + 12 }x(t)=t2+123000t​ t>0t > 0t>0 . Find the maximum value of x. Round your answer to two decimal places.

Find any critical numbers of the function g(t)=t14−tg ( t ) = t \sqrt { 14 - t }g(t)=t14−t​ , t < 14.

Locate the absolute extrema of the function f(x)=−4x2+8x−2f ( x ) = - 4 x ^ { 2 } + 8 x - 2f(x)=−4x2+8x−2 on the closed interval [−2,2][ - 2,2 ][−2,2] .

Find the open intervals on which the function y={2x+7,x≤74−x2,x>7y = \left\{ \begin{array} { l } 2 x + 7 , x \leq 7 \\4 - x ^ { 2 } , x > 7\end{array} \right.y={2x+7,x≤74−x2,x>7​ is increasing or decreasing.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

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Functions of Several Variables

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Find the x-value at which the given function has a point of inflection. $y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 15 x + 9$

Identify the open intervals where the function $f ( x ) = 4 x ^ { 2 } + 6 x - 1$ is increasing or decreasing.

Determine the open intervals on which the graph of $f ( x ) = 2 x ^ { 2 } + 5 x - 4$ is concave downward or concave upward.

Locate the absolute extrema of the given function on the closed interval [-48,48]. $f ( x ) = \frac { 48 x } { x ^ { 2 } + 16 }$

For the given function, find all critical numbers. $y = x ^ { 3 } - 3 x ^ { 2 } - 105 x + 5$

Find all relative minima of the given function. $y = x ^ { 4 } - 8 x ^ { 3 } + 16 x ^ { 2 } + 16$

Graph a function on the interval $[ - 1,5 ]$ having the following characteristics. Absolute maximum at $x = 4.75$ Absolute minimum at $x = - 1$ Relative maximum at $x = 0.3$ Relative minimum at $x = 3.1$

Locate the absolute extrema of the function $f ( x ) = x ^ { 3 } - 3 x$ on the closed interval [0,2].

Given $x y = 6,$ find $\frac { d y } { d t }$ when x = 4 and $\frac { d x } { d t } = - 2$

Find $y ^ { \prime }$ implicitly for $8 x ^ { 9 } - y ^ { 9 } = 5$

Find the third derivative. $y = \frac { 4 } { x ^ { 9 } }$

State the signs of $f ^ { \prime } ( x )$ and $f ^ { \prime \prime } ( x )$ on the interval (0, 2). $State the signs of f ^ { \prime } ( x ) and f ^ { \prime \prime } ( x ) on the interval (0, 2).$

Find the absolute extrema of the function $h ( t ) = ( t - 3 ) ^ { 2 / 3 }$ on the closed interval $[ - 1,6 ]$ . Round your answer to two decimal places.

Find the absolute extrema of the function $f ( x ) = \frac { 4 x } { x ^ { 2 } + 1 }$ on the interval $[ 0 , \infty )$ .

Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by $x ( t ) = \frac { 3000 t } { t ^ { 2 } + 12 }$ $t > 0$ . Find the maximum value of x. Round your answer to two decimal places.

Find any critical numbers of the function $g ( t ) = t \sqrt { 14 - t }$ , t < 14.

Locate the absolute extrema of the function $f ( x ) = - 4 x ^ { 2 } + 8 x - 2$ on the closed interval $[ - 2,2 ]$ .

Find the open intervals on which the function $y = \left\{ \begin{array} { l } 2 x + 7 , x \leq 7 \\4 - x ^ { 2 } , x > 7\end{array} \right.$ is increasing or decreasing.