Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Let a rectangle of length l and breadth b be inscribed in the given circle of radius a.

Then, the diagonal passes through the centre and is of length 2a cm.

Now, by applying the Pythagoras theorem, we have:

∴Area of the rectangle,

By the second derivative test, when, then the area of the rectangle is the maximum.

Since, the rectangle is a square.

Hence, it has been proved that of all the rectangles inscribed in the given fixed circle, the square has the maximum area.