Find the rectangle with the greatest area

Question

Find the rectangle with the greatest area, with a corner on O(0,0), one in the Ox axis and one on Oy axis, and one on the parabolic curve $y=4-x^2$ I thought to use the tangent of the parabolic function f'(x)=-2x. i don't know what to do next. For sides a and b the area is S=a*b

Answer 1

The opposite corner of $O(0,0)$ can be written as $P(t,4-t^2)$

So, we need to maximize $$f(t)=t(4-t^2)$$

Use Second Derivative test

Answer 2

You're right, it has something to do with derivatives, but you can't just scan the problem and look for the first function you see, and take the derivative of it.

This is an optimization problem, and we know that because of the word highest (it should probably be greatest). Greatest what? Greatest area. So construct the function that computes the area of the rectangle. That is the function you want to take the derivative of.