$$f(x,y)=x \log(y)$$
Partial derivatives: $$f_x(x,y)=\log(y)$$ $$f_y(x,y)=\frac{x}{y}$$
Stationary points:
\begin{cases} f_x(x,y)=\log(y)=0 \\ f_y(x,y)=\frac{x}{y}=0 \end{cases}
Solution: $(0,1)$
Hessian matrix in $(0,1)$:
$$f_{xx} (x,y)=0$$ $$f_{xy}(x,y)=f_{yx}(x,y)=\frac{1}{y}$$ $$f_{yy} (x,y)=-\frac{x}{y^2}$$
$$f_{xx} (0,1)=0$$ $$f_{xy}(0,1)=1$$ $$f_{yy} (0,1)=0$$
$$H_f(0,1)=\begin{pmatrix} 0 \ 1 \\ 1 \ 0 \end{pmatrix} $$
$$\det H_f(0,1)<0$$
$(0,1)$ is a Saddle point
Is it correct?
Thanks!