Find all points on he portion of the plane $x+y+z=5$ in the first octant at which $f(x,y,z)=xy^2z^2$ has a maximum value.
Attempt; Since $x+y+z=5$; $x=5-y-z$. I plug this into the $f(x,y,z)$: $f(5-y-z,y,z)=u(y,z)=y^2 z^2 (5-y-z)=\text{5 }y^2 z^2-y^3 z^2-y^2 z^3$
Now I find critical points: $u_y=10 yz^2-3y^2z^2-2yz^3=0$ $u_z=10 y^2 z-2 y^3 z-3 y^2 z^2=0$
The solution for this system of equations is $y=z=0$.Therefore, $x=5$. So thats the only critical point $(0,0,5)$ I get and $f(0,0,5)=0$. The answer should be a max at $(1,1,2)$ according to the answer key. How do I get it? Any hints please.