Proving legendre transform exists locally if $f$ twice differentiable, determinant of hessian of $f$ non-zero

Backgrounds

Legendre transform is defined as follows:

And the problem I am trying to solve follows:

My thoughts

I searched up to acquired that the matrix concerned is a hessian matrix.

Also, the previous subquestion has the following result:

Since this is from implicit function theorem section, I think it will be helpful, but I haven't worked my way around yet.

Any help is sincerely appreciated