Chapter 8

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=(x-5)^{2} $$

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. $$ f(x)=5-3 x $$

The radius \(r\) of a right circular cone is increasing at a rate of 2 inches per minute. The height \(h\) of the cone is related to the radius by \(h=3 r\). Find the rates of change of the volume when (a) \(r=6\) inches and (b) \(r=24\) inches.

In Exercises, find \(d y / d x\) $$ \frac{x y-y^{2}}{y-x}=1 $$

In Exercises, find the second derivative of the function. $$ f(x)=x \sqrt[3]{x} $$

In Exercises, find all relative extrema of the function. $$ f(x)=x^{4}-2 x^{3}+x+1 $$

A company that manufactures sport supplements calculates that its costs and revenue can be modeled by the equations \(C=125,000+0.75 x\) and \(R=250 x-\frac{1}{10} x^{2}\) where \(x\) is the number of units of sport supplements produced in 1 week. If production in one particular week is 1000 units and is increasing at a rate of 150 units per week, find: (a) the rate at which the cost is changing. (b) the rate at which the revenue is changing. (c) the rate at which the profit is changing.

In Exercises, find \(d y / d x\) $$ \frac{x+y}{2 x-y}=1 $$

In Exercises, find the second derivative of the function. $$ y=\left(x^{3}-2 x\right)^{4} $$

In Exercises, find all relative extrema of the function. $$ f(x)=x^{4}-12 x^{3} $$

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. $$ g(x)=-(x-1)^{2} $$

A company that manufactures pet toys calculates that its costs and revenue can be modeled by the equations \(C=75,000+1.05 x\) and \(R=500 x-\frac{x^{2}}{25}\) where \(x\) is the number of toys produced in 1 week. If production in one particular week is 5000 toys and is increasing at a rate of 250 toys per week, find: (a) the rate at which the cost is changing. (b) the rate at which the revenue is changing. (c) the rate at which the profit is changing.

In Exercises, find \(d y / d x\) $$ \frac{2 x+y}{x-5 y}=1 $$

In Exercises, find the second derivative of the function. $$ y=4\left(x^{2}+5 x\right)^{3} $$

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=x^{2 / 3}-3 $$

In Exercises, use a graphing utility to graph the function. Then find all relative extrema of the function. $$ f(x)=(x-1)^{2 / 3} $$

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. $$ y=x^{2}-6 x $$

All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter and (b) 10 centimeters?

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

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=x+\frac{4}{x} $$

In Exercises, use a graphing utility to graph the function. Then find all relative extrema of the function. $$ f(t)=(t-1)^{1 / 3} $$

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. $$ y=-x^{2}+2 x $$

Area All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the surface area changing when each edge is (a) 1 centimeter and (b) 10 centimeters?

In Exercises, find the second derivative of the function. $$ g(t)=-\frac{4}{(t+2)^{2}} $$

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

In Exercises, use a graphing utility to graph the function. Then find all relative extrema of the function. $$ g(t)=t-\frac{1}{2 t^{2}} $$

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. $$ y=x^{3}-6 x^{2} $$

A point is moving along the graph of \(y=x^{2}\) such that \(d x / d t\) is 2 centimeters per minute. Find \(d y / d t\) for each value of \(x\). (a) \(x=-3\) (b) \(x=0\) (c) \(x=1\) (d) \(x=3\)

In Exercises, find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. $$ y+x y=4 $$

In Exercises, find the second derivative of the function. $$ y=x^{2}\left(x^{2}+4 x+8\right) $$

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=\sqrt{2 x^{2}+6} $$

In Exercises, use a graphing utility to graph the function. Then find all relative extrema of the function. $$ f(x)=x+\frac{1}{x} $$

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. $$ y=(x-2)^{3} $$

A point is moving along the graph of \(y=1 /\left(1+x^{2}\right)\) such that \(d x / d t\) is 2 centimeters per minute. Find \(d y / d t\) for each value of \(x\). (a) \(x=-2\) (b) \(x=2\) (c) \(x=0\) (d) \(x=10\)

In Exercises, find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. $$ x^{3}-y^{2}=0 $$

In Exercises, find the second derivative of the function. $$ h(s)=s^{3}\left(s^{2}-2 s+1\right) $$

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=\sqrt{9-x^{2}} $$

In Exercises, use a graphing utility to graph the function. Then find all relative extrema of the function. $$ f(x)=\frac{x}{x+1} $$

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. $$ f(x)=\sqrt{x^{2}-1} $$

In Exercises, find the third derivative of the function. $$ f(x)=x^{5}-3 x^{4} $$

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=\sqrt{4-x^{2}} $$

In Exercises, use a graphing utility to graph the function. Then find all relative extrema of the function. $$ h(x)=\frac{4}{x^{2}+1} $$

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. $$ f(x)=\sqrt{9-x^{2}} $$

In Exercises, find the third derivative of the function. $$ f(x)=x^{4}-2 x^{3} $$

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=\frac{8}{x^{2}+2} $$

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

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. $$ y=x^{1 / 3}+1 $$

An air traffic controller spots two airplanes at the same altitude converging to a point as they fly at right angles to each other. One airplane is 150 miles from the point and has a speed of 450 miles per hour. The other is 200 miles from the point and has a speed of 600 miles per hour. (a) At what rate is the distance between the planes changing? (b) How much time does the controller have to get one of the airplanes on a different flight path?

In Exercises, find the third derivative of the function. $$ f(x)=5 x(x+4)^{3} $$

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=\frac{18}{x^{2}+3} $$