Convergence Behavior at a Saddle Point of the Cost Surface

You are minimizing a smooth scalar cost $f(x)$ over $x \in \mathbb{R}^n$ (for example, an energy/objective in motion planning or learned-controller training). At a point $x^\star$ you find that the gradient $\nabla f(x^\star) = 0$ and the Hessian $\nabla^2 f(x^\star)$ has both strictly positive and strictly negative eigenvalues.

Which statement is correct about $x^\star$ and the behavior of optimization algorithms there?