Answer
$(\dfrac{100}{3} ,\dfrac{100}{3} ,\dfrac{100}{3})$
Work Step by Step
1. If $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \gt 0$ and $f_{xx}(p,q)\gt 0$ ,
then $f(p,q)$ is a local minimum.
2.If $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \gt 0$ and $f_{xx}(p,q)\lt 0$ , then
$f(p,q)$ is a local maximum.
3. If $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \lt 0$ , then $f(p,q)$ is a not a
local minimum and local maximum or, a saddle point.
From the given problem $z=100-x-y$
$D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \gt 0$ and $f_{xx}(p,q)\lt 0$ , then $f(p,q)$ is a local maximum.
$D(\dfrac{100}{3},\dfrac{100}{3})=\dfrac{200}{3} \gt 0$ and $f_{xx}=-\dfrac{200}{3} \lt 0$
The required points are: $(\dfrac{100}{3} ,\dfrac{100}{3} ,\dfrac{100}{3})$