Optimization Problem

Maximize the product of two numbers, x and y, given that the first number squared added to the second number is equal to 27.

Find two non-negative numbers, x and y, whose sum is 9 with a maximum product of x and the square of y, $p(x,y) = xy^2$

Find the maximum value of $f(x) = xe^{-x}$ on $(-\infty, \infty)$

A company makes wrenches. It costs them an initial \$10,000 to set up and an additional \$5.00 for each wrench. A market study indicates that if the price of a wrench is \$5.00 then the company will sell 200,000 wrenches. However, for every \$10 that the company raises the price per wrench, only half as many people will buy the wrench. What price should they charge per wrench to maximize the companies profits.