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.


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