91Ó°ÊÓ

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

Show that the rectangle of fixed area whose perimeter is a minimum is a square.

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

Sketch the graph of each function "by hand" after making a sign diagram for the derivative and finding all open intervals of increase and decrease. \(f(x)=x^{4}+4 x^{3}-8 x^{2}+64\)

ENVIRONMENTAL SCIENCE: Maximum Sustainable Yield The reproduction function for the Pacific sardine (from Baja California to British Columbia) is \(f(p)=\) \(-0.0005 p^{2}+2 p\), where \(p\) and \(f(p)\) are in hundred metric tons. Find the population that gives the maximum sustainable yield, and the size of the yield.

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

Find (without using a calculator) the absolute extreme values of each function on the given interval. $$ f(x)=\frac{x}{x^{2}+1} \text { on }[-3,3] $$

Sketch the graph of each function "by hand" after making a sign diagram for the derivative and finding all open intervals of increase and decrease. \(f(x)=x^{4}-4 x^{3}-8 x^{2}+64\)

Find (without using a calculator) the absolute extreme values of each function on the given interval. $$ f(x)=\frac{1}{x^{2}+1} \text { on }[-3,3] $$

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