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
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