Suppose a function $f$ from $\mathbb{R}^n \to \mathbb{R}$, is differentiable. We know that $c$ is a critical point of $f$, i.e. $\nabla f(c) = 0$. Our goal is to find out if $c$ is a local extremum, or a saddle point, without computing the Hessian.
Does this suggestion work?
Take $n$ orthogonal vectors (directions) $v$ in $\mathbb{R}^n$. Examine $f$ at $c$: If all $n$ vectors $v$s are ascent directions, we are at a local minimum, if they are all descent directions we are at a local maximum, otherwise we are at a saddle point.