indefinite system for finding extreme values

Question

The function $f(x,y) = xe^y$ has a false system does this mean that there are no extreme values?

EDIT: false system not an indefinite system sorry for the confusion

Answer 1

Not sure what an indefinite system means in this context, but since $f(x,y) = g(x)h(y)$ your maximum occurs when both $g,h$ have their respective maxima. For the minimum, note that $g(x)$ and $h(y)$ are both unbounded as $x,y \to \infty$. Also, $h(y)=e^y \ge 0$, so let $(x,y) \to (-\infty,\infty)$.

UPDATE

You are right, to find the stationary points you have to solve $$ \begin{pmatrix} 0 \\ 0 \end{pmatrix} = \vec{\nabla} f(x,y) = \begin{pmatrix} e^y \\ xe^y \end{pmatrix} $$ and the first equation has no solutions. Therefore, there are no stationary points of $f(x,y)$. The reason for this can be explained like the original argument above.