Is this Calculus? Let me assume I can use Calculus.

I understand that you mean, "Demonstate that in all rectangles that have the same perimeters, the square has the biggest area."

Perimeter, P = 2(L +w) ---------(1)

Area, A = w*L ---------------(2)

From (1),

L +w = P/2

L = P/2 -w

So,

A = w(P/2 -w)

A = (P/2)w -w^2

Differentiate both sides with respect to w,

dA/dw = P/2 -2w

Set that to zero,

0 = P/2 -2w

2w = P/2

4w = P ----------***

Substitute that into (1),

P = 2(L +w)

4w = 2L +2w

4w -2w = 2L

2w = 2L

L = w ---------meaning, length = width, which is a square.

Therefore, the rectangle that is a square has the biggest area.