Consider a right triangle with hypotenuse 1. Let and denote the lengths of the legs of the triangle. Find the values of and such that the value is maximized.

We have the following right triangle:

From the Pythagorean theorem we have

So we want to maximize,

Calling this function and taking the derivative,

Setting this equal to 0 we find the critical points of the function ,

Since

we have that has a maximum when . Therefore, since we then have

Thus, the maximum of is