Let one side be x and other be y. Then 2x+2y=40 if the perimeter is 40. We want to find the maximum of f(x,y)=xy given perimeter is 40. We can decrease the number of unknowns by substitution using the perimeter equation.

2y+2x=40, y=20-x

f(x,y)=g(x)=x(20-x)=20x-x² will be the function of one variable to be maximized.

g_x=20-2x=0 then x=10, since y=20-x=10. So the area will be 10*10=100