Exam 4: Extrema on an Interval

Suppose a manufacturer has determined that the total cost C of operating a factory is $C = 0.6 x ^ { 2 } + 14 x + 54,000$ where $x$ is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost per unit is $\mathrm { C } / x$ .)

$\text { Determine whether Rolle's Theorem can be applied to } f ( x ) = - x ^ { 2 } + 10 x \text { on the closed }$ interval $[ 0,10 ]$ . If Rolle's Theorem can be applied, find all values of $c$ in the open interval $( 0,10 )$ such that $f ^ { t } ( c ) = 0$

Locate any relative extrema and inflection points of the function $y = \frac { x ^ { 15 } } { 15 } - \ln x$ . Use a graphing utility to confirm your results.

$\text { For the function } f ( x ) = 4 x ^ { 3 } - 48 x ^ { 2 } + 6 \text { : }$ (a)Find the critical numbers of $f$ (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. Then use a graphing utility to confirm your results.

Determine the x-coordinate(s) of any relative extrema and inflection points of the function $y = x ^ { 5 } \ln \frac { x } { 9 }$

Find the points of inflection and discuss the concavity of the function $f ( x ) = - 8 x - 2 \cos x$ on the interval $[ 0,2 \pi ]$

Analyze and sketch a graph of the function $y = x ^ { 3 } - 3 x ^ { 2 } + 3$

Find the length and width of a rectangle that has an area of 968 square feet and whose perimeter is a minimum.

A plane begins its takeoff at 2:00 P.M. on a 2200-mile flight. After 12.5 hours, the plane arrives at its destination. Explain why there are at least two times during the flight when the Speed of the plane is 100 miles per hour.

Match the function $f ( x ) = 6 - \frac { 6 x } { \sqrt { x ^ { 2 } + 2 } }$ , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x 1 1 1 1 1 1 1 f(x)

Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the Total perimeter is 38 feet.

Find all points of inflection on the graph of the function $f ( x ) = \frac { 1 } { 2 } x ^ { 4 } + 2 x ^ { 3 }$

Analyze and sketch a graph of the function $y = 2 - 3 x - x ^ { 3 }$

Find the differential dy of the function $y = 2 x ^ { 3 / 7 }$

Find the point on the graph of the function $f ( x ) = ( x + 1 ) ^ { 2 }$ that is closest to the point $( - 5,1 )$ . Round all numerical values in your answer to four decimal places.

$\text { Find the cubic function of the form } f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d \text {, where } a \neq 0 \text { and the }$ coefficients $a , b , c , d$ are real numbers, which satisfies the conditions given below. Relative maximum: $( 3,0 )$ Relative minimum: $( 5 , - 2 )$ Inflection point: $( 4 , - 1 )$

A rectangle is bounded by the x- and y-axes and the graph of $y = \frac { ( 5 - x ) } { 2 }$ figure). What length and width should the rectangle have so that its area is a maximum?

$\text { Use the graph of the function } y = \frac { x ^ { 3 } } { 4 } - 3 x \text { given below to estimate the open }$ intervals on which the function is increasing or decreasing.

$\text { Locate the absolute extrema of the function } g ( t ) = \frac { t ^ { 2 } } { t ^ { 2 } + 2 } \text { on the closed interval }$ $[ - 3,3 ] \text {. }$

$\text { Locate the absolute extrema of the given function on the closed interval } [ - 60,60 ] \text {. }$ $f ( x ) = \frac { 60 x } { x ^ { 2 } + 36 }$