Exam 3: Additional Applications of the Derivative

Find the absolute maximum and absolute minimum of $f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 2$ on the interval -2 $\le$ x $\le$ 5.

Find all the critical numbers of the function $f ( x ) = x ^ { 3 } - 3 x - 2$ .

The inflection point of $f ( x ) = x ^ { 3 } + 6 x ^ { 2 } - 13$ is (-2, 3).

Determine where the graph of $f ( x ) = x ^ { 4 } - 6 x ^ { 2 } + 2$ is concave up.

= The absolute maximum of the function $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 3$ on the interval -1 $\le$ x $\le$ 2 is 11.

Use the second derivative test to find the relative maxima and minima of the function $f ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 108 x + 1$

The cost of producing x units of a certain commodity is $C ( x ) = 3 x ^ { 2 } + 4 x + 7$ dollars. If the price is p(x) = (49 - x) dollars per unit, determine the level of production that maximizes profit.

Find two positive numbers whose sum is 100 and whose product is as large as possible.

If the cost of a commodity is $C ( x ) = \frac { 1 } { 8 } x ^ { 2 } + 3 x + 98$ dollars when x units are produced and the selling price is $p ( x ) = \frac { 1 } { 3 } ( 75 - x )$ dollars per unit, find the level of production where profit is maximized.

The owner of an appliance store expects to sell 700 toasters this year. Each toaster costs her $8 dollars to purchase, and each time she orders a shipment of toasters, it costs $34. In addition, it costs $3 a year to store each toaster. Assuming the toasters sell out at a uniform rate and that the owner never allows herself to run out of toasters, how many toasters should be ordered in each shipment to minimize the annual cost? (Round any fractional amounts.)

A poster is to contain 108 cm² of printed matter with additional margins 6 cm each at top and bottom and 2 cm on the sides. What is the minimal cost of the poster if it is to be made out of material costing 20 cents per square centimeter?

A store uses 900 cases of electronic parts each year. The cost of storing one case for a year is 80 cents and the ordering fee is $40 per shipment. How many cases should the store order each time to minimize total cost? Assume the orders are planned so that a new shipment arrives just as the number of cases in the store reaches zero. Also assume the parts are consumed at a constant rate.

The owner of a novelty store can obtain joy buzzers from the manufacturer for 30 cents apiece. He estimates he can sell 40 buzzers when he charges $1.40 apiece for them and that he will be able to sell 8 more buzzers for every 8 cent decrease in price. What price should he charge in order to maximize profit?

The absolute minimum of the function $f ( x ) = 8 x ^ { 3 } - 4 x ^ { 2 } + 72 x$ on the interval 0 $\le$ x $\le$ 4 is 0.

Find the intervals of increase and decrease for the function $f ( x ) = - x ^ { 7 } - x ^ { 5 } - 3 x + 15$ .

Determine where the graph of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 1$ is concave down.

The revenue derived from the production of x units of a particular commodity is $R ( x ) = \frac { 15 x - x ^ { 2 } } { x ^ { 2 } + 15 }$ million dollars. What level of production results in maximum revenue? What is the maximum revenue? Round numbers to two decimal places, if necessary.

Find constants a, b, and c so that the graph of the function $f ( x ) = a x ^ { 2 } + b x + c$ has a relative maximum at (5, 17) and crosses the y-axis at (0, 7).

Find the elasticity n of the demand function $D ( p ) = \frac { 3 } { 1 + 2 p ^ { 2 } }$ .

Determine where the graph of $f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 6 x + 3$ is concave down.