Find the dimensions of the rectangular box with largest volume if the total surface area is given as $64$cm$^2$
$$ v=f(x,y,z)= xyz\\ 2xy+2yz+2xz = 64\\ xy+yz+xz = 32 $$
Then I solved for $xy$: $$ xy = 32 - yz - xz $$ Substituted back in $v$: $$ f(x,y,z) = z(32 - yz - xz) = 32z - yz^2-xz^2 $$ Then, I tried to find critical points for that function, to find where I could get the maximum volume: $$ f_x = -z^2\\ f_y = -z^2\\ f_z = 32-2zy-2zx\\ $$
And now I'm lost, because if z=0 from $f_x$ and $f_y$, I can't get $f_z=0$...
What have I done wrong? Do I need to solve for just one variable and then substitute? What's going on here? I really don't get why it's "incorrect" to solve for two variables since my "main" function (in this case $v$) has these two variables in it...
EDIT: $$ xy = 32z - yz - xz $$
is not solved in terms of $xy$, since my right side of equation still has x and y... that's what I have done wrong?!