Exam 3: Additional Applications of the Derivative

Sketch the graph of the given function. $f ( x ) = \frac { x - 5 } { x - 3 }$

Which non-negative number exceeds its own fourth power by the greatest amount?

The absolute minimum of $f ( x ) = \frac { 9 } { x } + x - 3$ on the interval 1 $\le$ x $\le$ 9 is 1.

Determine where the graph of $f ( x ) = \frac { x ^ { 2 } } { 7 }$ is concave up and concave down.

The function graphed below has a positive second derivative everywhere.

The owner of a novelty store can obtain joy buzzers from the manufacturer for 70 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 15 more buzzers for every 10 cent decrease in price. What price should he charge in order to maximize profit?

Find the intervals of increase and decrease for the function $f ( x ) = \sqrt { x ^ { 6 } + 8 }$ .

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval 1 $\le$ x $\le$ 7.

Find the absolute maximum of the function $f ( x ) = x ^ { 3 }$ on the interval $- \frac { 1 } { 2 } \leq x \leq 1$ .

Find the intervals of increase and decrease for the function $f ( x ) = - x ^ { 5 } - x ^ { 3 } - 4 x + 10$ .

Determine where the graph of $f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 7 x + 5$ is concave down.

Determine where the graph of $g ( x ) = \frac { 1 } { x ^ { 2 } + 3 }$ is concave up and concave down.

The second derivative test reveals that $f ( x ) = x ^ { 4 } - 4 x ^ { 2 } + 1$ has

The cost of producing x units of a certain commodity is $C ( x ) = 3 x ^ { 2 } + 4 x + 8$ dollars. If the price is p(x) = (50 - x) dollars per unit, determine the level of production that maximizes profit.

Find the absolute maximum and minimum of the function $f ( x ) = \frac { 1 } { 6 } \left( x ^ { 3 } - 6 x ^ { 2 } + 9 x + 1 \right)$ on the interval 0 $\le$ x $\le$ 2.

To raise money, a service club has been collecting used bottles that it plans to deliver to a local glass company for recycling. Since the project began 40 days ago, the club has collected 20,000 pounds of glass for which the glass company currently offers 1 cent per pound. However, because bottles are accumulating faster than they can be recycled, the company plans to reduce by 1 cent each day the price it will pay for 100 pounds of used glass. Assume that the club can continue to collect bottles at the same rate and that transportation costs make more than one trip to the glass company unfeasible. What is the most advantageous time for the club to conclude its project and deliver the bottles?

If the total cost of manufacturing a commodity is $C ( x ) = \frac { 1 } { 8 } x ^ { 2 } + 4 x + 200$ dollars when x units are produced, for what value of x is the average cost the least?

Find all the critical numbers of the function $f ( x ) = - 2 x ^ { 4 } + 4 x ^ { 2 } + 1$ .

A house is located in the desert, 13 miles from a long, straight road. A man in a town 18 miles down the road arranges to be transported to the house by a company that charges $4 per mile on the road and $6 per mile on the desert. What is the minimum cost of the trip?

The second derivative test reveals that f (x) =x²- 5 has