91Ó°ÊÓ

For each equation, use implicit differentiation to find \(d y / d x\). $$ x y-x=9 $$

The U.S. Postal Service will accept a package if its length plus its girth (the distance all the way around) does not exceed 84 inches. Find the dimensions and volume of the largest package with a square base that can be mailed.

For each function: a. Make a sign diagram for the first derivative. b. Make a sign diagram for the second derivative. c. Sketch the graph by hand, showing all relative extreme points and inflection points. $$ f(x)=x^{4}-8 x^{3}+18 x^{2}+2 $$

Find the critical numbers of each function. \(f(x)=(2 x-6)^{4}\)

Find the critical numbers of each function. \(f(x)=\left(x^{2}+6 x-7\right)^{2}\)

For each equation, use implicit differentiation to find \(d y / d x\). $$ x^{3}+2 x y^{2}+y^{3}=1 $$

ENVIRONMENTAL SCIENCE: Maximum Sustainable Yield The reproduction function for the Hudson Bay lynx is estimated to be \(f(p)=-0.02 p^{2}+5 p\), where \(p\) and \(f(p)\) are in thousands. Find the population that gives the maximum sustainable yield, and the size of the yield.

A homeowner wants to build, along his driveway, a garden surrounded by a fence. If the garden is to be 800 square feet, and the fence along the driveway costs \(\$ 6\) per foot while on the other three sides it costs only \(\$ 2\) per foot, find the dimensions that will minimize the cost. Also find the minimum cost.

Find (without using a calculator) the absolute extreme values of each function on the given interval. $$ f(x)=6 x^{2}-x^{3} \text { on }[0,5] $$

For each function: a. Make a sign diagram for the first derivative. b. Make a sign diagram for the second derivative. c. Sketch the graph by hand, showing all relative extreme points and inflection points. $$ f(x)=x^{4}+8 x^{3}+18 x^{2}+8 $$