Showing existence of solution by positive definiteness/convexity

Showing existence of solution by positive definiteness/convexity

For a physics problem, I am considering the following problem: I have a
certain function, $S: \mathbb{R}^M \rightarrow \mathbb{R}$, of which the
critical points, given by $$ \frac{\partial S}{\partial\lambda_{\gamma}}=0
$$ ($\lambda\in\mathbb{R}^M$ so M equations, I denote a component of this
vector by $\lambda_{\gamma}$). In fact, the critical points form an
equation I am actually interested in. I want to prove that these equations
have a unique, existing solution. The reference I am following, shows that
$$ \sum_{\alpha,\beta}v_{\alpha}v_{\beta}\frac{\partial^2
S}{\partial\lambda_{\alpha}\partial\lambda_{\beta}}>0¸ $$ i.e. S is
positive definite in lambda. It then concludes that the equations thus
have a unique, existing solution.
I am somewhat confused by this. I know of a theorem that states that if
you have a local minimum of a function and that function is strictly
convex ($\Leftrightarrow$ it is positive definite), then the local minimum
is in fact a global minimum. But in this case we only know that the
equations are an extrema of S. Can anyone shed his/her light on this?