Calculus Optimization

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

A dudebro has 500 feet of fencing to make a pig pen along a barn as shown below. Find the maximum area.

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 $(-\infty, \infty)$