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).