Maximum Volume (Dimensions of Box)

Question

$$V(x, y) = xy(96-x-y)$$

Why is one of the critical points equal to $(32, 32)$ when we take the partial derivatives we get $$V_x = y(96-2x-y)= 0 \text{ and } V_y= x(96-2y-x)= 0.$$ But when I solve for the critical points I get $(0,0)$, $(96, 0)$, $(0, 96)$ but not $(32, 32)$. Can anyone explain why?

Answer 1

Any solution of $V_x=V_y=0$ with $x,\,y$ nonzero satisfies $x+2y=2x+y=96$, i.e. $x=y=32$.