Given a square with edges each having length , find the area of the largest square that can be circumscribed about the given square.

Let be the length of the edge of the circumscribed square. Then . Furthermore, if is the length of the edge of the given square, we have

Then,

Let this function be , and take the derivative,

Thus, the derivative is zero when . Furthermore,

Therefore, is increasing when and decreasing when ; hence, has a maximum at . Solving for we then have

Finally, since we then have the area of the circumscribed square given by

The dirivative seems to be wrong. the second term should be – x^2 / sqrt(L^2 – x^2).