Chapter 8

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ f(x)=\frac{1}{3}(2 x+5), \quad[0,5] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ y=x^{2 / 3}-4 $$

In Exercises, find the third derivative of the function. $$ f(x)=\left(x^{3}-6\right)^{4} $$

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=\frac{x}{x-1} $$

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ f(x)=5-2 x^{2}, \quad[0,3] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ g(x)=(x-1)^{1 / 3} $$

A (square) baseball diamond has sides that are 90 feet long (see figure). A player 26 feet from third base is running at a speed of 30 feet per second. At what rate is the player's distance from home plate changing?

In Exercises, find the third derivative of the function. $$ f(x)=\frac{3}{16 x^{2}} $$

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=\frac{x}{x^{2}-1} $$

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ f(x)=x^{2}+2 x-4, \quad[-1,1] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ g(x)=(x-1)^{2 / 3} $$

A retail sporting goods store estimates that weekly sales \(S\) and weekly advertising costs \(x\) are related by the equation \(S=2250+50 x+0.35 x^{2}\). The current weekly advertising costs are \(\$ 1500\), and these costs are increasing at a rate of \(\$ 125\) per week. Find the current rate of change of weekly sales.

In Exercises, find the third derivative of the function. $$ f(x)=\frac{1}{x} $$

In Exercise, use a graphing utility to estimate graphically all relative extrema of the function. $$ f(x)=\frac{1}{2} x^{4}-\frac{1}{3} x^{3}-\frac{1}{2} x^{2} $$

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ f(x)=x^{3}-3 x^{2}, \quad[-1,3] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=-2 x^{2}+4 x+3 $$

An accident at an oil drilling platform is causing a circular oil slick. The slick is \(0.08\) foot thick, and when the radius of the slick is 150 feet, the radius is increasing at the rate of \(0.5\) foot per minute. At what rate (in cubic feet per minute) is oil flowing from the site of the accident?

In Exercise, use a graphing utility to estimate graphically all relative extrema of the function. $$ f(x)=-\frac{1}{3} x^{5}-\frac{1}{2} x^{4}+x $$

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ f(x)=x^{3}-12 x, \quad[0,4] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=x^{2}+8 x+10 $$

A company is increasing the production of a product at the rate of 25 units per week. The demand and cost functions for the product are given by \(p=50-0.01 x\) and \(C=4000+40 x-0.02 x^{2} .\) Find the rate of change of the profit with respect to time when the weekly sales are \(x=800\) units. Use a graphing utility to graph the profit function, and use the zoom and trace features of the graphing utility to verify your result.

In Exercises, find the given value. $$ f(x)=9-x^{2} \quad f^{\prime \prime}(-\sqrt{5}) $$

In Exercise, use a graphing utility to estimate graphically all relative extrema of the function. $$ f(x)=5+3 x^{2}-x^{3} $$

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ h(s)=\frac{1}{3-s}, \quad[0,2] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ y=3 x^{3}+12 x^{2}+15 x $$

The profit for a product is increasing at a rate of \(\$ 5600\) per week. The demand and cost functions for the product are given by \(p=6000-25 x\) and \(C=2400 x\) + 5200 . Find the rate of change of sales with respect to time when the weekly sales are \(x=44\) units.

In Exercise, use a graphing utility to estimate graphically all relative extrema of the function. $$ f(x)=3 x^{3}+5 x^{2}-2 $$

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ h(t)=\frac{t}{t-2}, \quad[3,5] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ y=x^{3}-3 x+2 $$

The annual cost (in millions of dollars) for a government agency to seize \(p \%\) of an illegal drug is given by \(C=\frac{528 p}{100-p}, 0 \leq p<100\) The agency's goal is to increase \(p\) by \(5 \%\) per year. Find the rates of change of the cost when (a) \(p=30 \%\) and (b) \(p=60 \%\). Use a graphing utility to graph \(C\). What happens to the graph of \(C\) as \(p\) approaches \(100 ?\)

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ f(x)=3 x^{2 / 3}-2 x, \quad[-1,2] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=x \sqrt{x+1} $$

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ g(t)=\frac{t^{2}}{t^{2}+3}, \quad[-1,1] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ h(x)=x \sqrt[3]{x-1} $$

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ h(t)=(t-1)^{2 / 3}, \quad[-7,2] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=x^{4}-2 x^{3} $$

In Exercises, find the higher-order derivative. $$ f^{\prime}(x)=2 x^{2} $$

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ g(x)=4\left(1+\frac{1}{x}+\frac{1}{x^{2}}\right), \quad[-4,5] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=\frac{1}{i} x^{4}-2 x^{2} $$

In Exercises, find the higher-order derivative. $$ f^{\prime \prime}(x)=20 x^{3}-36 x^{2} $$

In Exercises, find the point(s) of inflection of the graph of the function. $$ f(x)=x^{3}-9 x^{2}+24 x-18 $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=\frac{x}{x^{2}+4} $$

In Exercises, find the higher-order derivative. $$ f^{\prime \prime \prime}(x)=(3 x-1) / x $$

In Exercises, find the point(s) of inflection of the graph of the function. $$ f(x)=x(6-x)^{2} $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=\frac{x^{2}}{x^{2}+4} $$

In Exercises, find the point(s) of inflection of the graph of the function. $$ f(x)=(x-1)^{3}(x-5) $$

In Exercises, use a graphing utility to find graphically the absolute extrema of the function on the closed interval. $$ f(x)=0.4 x^{3}-1.8 x^{2}+x-3, \quad[0,5] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. (Hint: Check for discontinuities.) Sketch the graph of the function. $$ f(x)=\frac{2 x}{16-x^{2}} $$

In Exercises, find the point(s) of inflection of the graph of the function. $$ f(x)=x^{4}-18 x^{2}+5 $$

In Exercises, use a graphing utility to find graphically the absolute extrema of the function on the closed interval. $$ f(x)=3.2 x^{5}+5 x^{3}-3.5 x, \quad[0,1] $$