Let $(M^n,g), n \geq 3$ be a closed Riemannian manifold. Assume that there is a function $\phi : M \to \mathbb{R}$ of class $C^2$ such that $\phi$ has a saddle point. Then, is necessarily true that in this point $M$ has negative scalar curvature?

It seems true, but I just have an intuition about it, like saddle points have a two plane with negative sectional curvature...