Given a function $f:\Bbb R^2 \to \Bbb R, f \in C^\infty$ with the property that
$$\lim_{x\to+\infty}f(x,y_0) = \lim_{x\to-\infty}f(x,y_0) = +\infty \qquad \forall y_0\in \Bbb R, \\[2ex]
\lim_{y\to+\infty}f(x_0,y) = \lim_{y\to-\infty}f(x_0,y) = -\infty \qquad \forall x_0\in \Bbb R.$$
Determine whether $f(x,y)$ necessarily has at least one critic point.
My attempt:
I suppose that such a function could be something just like: $(x^{2n}-y^{2m})$; anyway what I mean is that both $f(x,y_0)$ and $f(x_0,y)$ have to assume eventually the shape of a sort of "parabola".
Because of $f\in C^{\infty}$ then both $f(x,y_0)$ and $f(x_0,y)$ are continuous, thus:
1.$f(x,y_0)=g(x),$ has a global minimum and it means: $\forall y_0 \in \Bbb R$ there is at least one $x^*$ such that $f_x(x^*,y_0)=0;$
2.$f(x_0,y)=h(y),$ has a global maximum and it means: $\forall x_0 \in \Bbb R$ there is at least one $y^*$ such that $f_y(x_0,y^*)=0.$
If my attempt is correct until now, the last thing I need to do is observe that there is at least a couple $(x^*,y^*)$ such that $f_x(x^*,y^*)=0$ and $f_y(x^*,y^*)=0$. This last step is the one I stuck in.
Is there someone who can handle this? (or can propose another path to follow)