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