To find hottest point on probe surface A space probe in form of ellipsoid $4x^2+y^2+4z^2=16$ enters earth atmosphere and its surface begins to heat. After one hour, temperature at point $(x,y,z)$ on probe surface is given by $$T(x,y,z) = 8x^2+4yz-16z+1600.$$ Find hottest point on the surface probe. I computed gradient of $T$ but don't know how to proceed next. Thanks for help.

$$T(x,y,z)\leq8x^2+2(y^2+z^2)-16z+1600=$$ $$=8x^2+2y^2+8z^2-6z^2-16z+1600=$$ $$=-6z^2-16z+1632=-\frac{2}{3}(3z+4)^2+\frac{4928}{3}\leq\frac{4928}{3}.$$ The equality occurs for $(x,y,z)=\left(\pm\frac{4}{3},-\frac{4}{3},-\frac{4}{3}\right)$ only, which says that we got a maximal value and two needed points.