Show that of all the rectangles with a given perimeter, the square has the largest area.

Let us Consider a rectangle with length is x cm and breadth is y cm.

Since, Area of rectangle = length × breadth

A=xy …(i)

And, Perimeter of rectangle is , = 2(length + breadth)

P=2(x + y) …(ii)

Now, From (ii),

Putting the value of y in equation (ii), we get

Now, differentiate A, w.r.t x , we get

If then

If

Since,

Corresponds to a maximum value of A.

The area is maximum when the sides are

Hence, This is the square, with largest area.

Hence, Proved