Question 1087451
Max area is when the rectangle is a square:  L=W= {{{ highlight(1/4) }}}
—

Proof:   2L + 2W = 1  —> {{{ W=(1-2L)/2 }}}
             A = LW = {{{ L(1-2L)/2  = (1/2)(L-2L^2) }}}

Take derivative of A with respect to L:
             dA/dL = {{{ (1/2)(1-4L) }}}

Set that to 0 and solve for L, while noting that {{{d^2A/dL^2 = -2 }}} —> Concave down so our answer will give us a MAX:
                  {{{ (1/2)(1-4L) = 0 }}}
                  {{{   4L = 1 }}}
                   {{{  L = 1/4 }}}  —>  {{{ W = 1/4 }}}   
 
              Since L=W we have a square.