# proof of second derivative test for function of n variables

Assume that $$f : \mathbb{R^n}\to\mathbb{R}$$ is a $$C^2$$ function and that $$\mathbf{a}$$ is a point such that $$∇f(\mathbf{a}) = 0$$. Assume also that $$H(\mathbf{a})$$ has at least one negative eigenvalue $$−λ > 0$$, with an eigenvector $$\mathbf{v}$$ (normalized so that $$|\mathbf{v}| = 1$$. Prove that there exists $$s_0 > 0$$ such that $$f(\mathbf{a}+s\mathbf{v}) < f(\mathbf{a}) - \frac14\lambda s^2$$ for $$|s| < s_0$$.

I was given this question to solve for the proof of the necessary condition of second derivative test. I've tried using the Taylor's theorem to simplify the inequality; however, I am stuck with finding the relationship between the Hessian matrix, the significance the the negative eigenvalue, and how the unit vector $$\mathbf{v}$$ plays a role in solving the inequality.