Multivariable local maximum proof

Question

Suppose we have a twice differentiable function $f: \mathbb{R} ^n \to \mathbb{R}$, a point ${\bf x^0} = (x_1 ^0 , \ldots , x_n ^0)$ and we know that

1. $\nabla f({\bf x}^0) = 0$
2. $({\bf x - x^0})H({\bf x^0})({\bf x - x^0})^T <0 $, $(H({\bf x^0})$ is the Hessian matrix of $f$)

Prove that $f({\bf x^0})$ is the local maximum of $f$.

Attempt at a solution

Since $\nabla f({\bf x}^0) = 0$ we know that ${\bf x^0}$ is a critical point of the function $f$. Because the Hessian is smaller than $0$ at the point ${\bf x^0}$, $f$ has a local maximum at the point ${\bf x^0}$.$\hspace{0.5em} \square$

Here is my dilemma. My "proof" feels very cheap. However, I'm not taking a proof heavy class (in fact proofs of why $f$ has a local maximum at a critical point if the Hessian is smaller than zero are not included in my text!) so I don't know what to add ... I feel like I have said everything that has to be said so I'm hoping any of you can point out to me how I can buff it up. I just want to add that this question isn't really homework, it's from an old final by my teacher which I'm solving in my free time. Thanks.

Improved solution

We know that $\nabla f({\bf x}^0) = 0$. Then, by definition, ${\bf x}^0$ is a critical point of $f$. Define function $g({\bf x}^0) = ({\bf x - x^0})H({\bf x^0})({\bf x - x^0})^T < 0$. Since $g({\bf x}^0) < 0$, all the eigenvalues of $H({\bf x^0})$ are less than $0$ resulting in $f$ having a local maximum at the point ${\bf x^0}$.

All there is left to do is reference where I use known theorems. Is this sufficient?

Answer 1

Property 2 is a statement about a matrix, but in your proof, you say 'the Hessian is smaller than $0$'. This doesn't quite follow.

If you want some details to fill out, you can start there - assuming you know that "a critical point of $f$ is a maximum if the determinant of the Hessian is smaller than zero" how can you connect property 2 to a negative determinant?

If you really wanted to tear this apart, you could start from scratch, and prove "a critical point of $f$ is a maximum if the determinant of the Hessian is smaller than zero".