Question 894736
Length is L, width is w.


{{{A=wL}}}, and {{{2w+2L=4}}}, the constant perimeter of 4 feet.
Simplify perimeter equation to {{{w+L=2}}}.


Part (a) asks for A(L).
from perimeter equation, w=2-L.
{{{A=(2-L)L}}}
{{{A=2L-L^2}}}
Or if  you want, 
{{{highlight(A(L)=2L-L^2)}}}, area as a function of length.


The function A(L) is quadratic, and has a maximum value, based on coefficient on {{{L^2}}} being {{{-1}}}.


The maximum value for area will be at L=w, meaning a square shape.  This will simply be L=1 and w=1; and the area will be 1 square foot.


Part (b): you asked for an algebraic explanation but I did not give it that way.  I simply gave a discussion.


Try this:
Find roots for A(L).
{{{2L-L^2=0}}}
{{{L(2-L)=0}}}
{{{L=0}}} or {{{L=2}}}.
The maximum value occurs in the exact middle of these roots.