Let $x_0$ and $y_0$ - stationary points of twice differentiable functions $f(x)$ and $g(y)$.
And $f(x_0)=0$.
Find sufficent condition for $g(y)$ so that function $z(x,y)=f^2(x)\cdot g(y)$ has an extremum at the point $(x_0,y_0)$.
First I found necessary condition - $z^{'}_x=2f(x)f^{'}(x)g(y)=0$ ; $z^{'}_y=f^2(x)g^{'}(y)=0$.
Both conditions are met at point $(x_0,y_0)$.
Sufficent condition - $\Delta =\begin{vmatrix} z^{''}_{xx} &z^{''}_{xy} \\ z^{''}_{yx}& z^{''}_{yy} \end{vmatrix}> 0$ . But $\Delta =0$...