When we are dealing with multivariate functions of type $f(\vec x)$ (which has a critical point at let's say $\vec x_o$), we find whether it's a minima or maxima by writing the Taylor's expansion of $f$ which is of the form:
$$f(\vec x) = f(\vec x_o) + \nabla f(\vec x_o) \cdot (\vec x - \vec x_o) + 1/2 (\vec x - \vec x_o)^TH_f(\vec x - \vec x_o) + ...$$
Then, we say that if $H$ is positive definite, it has a minima. If $H$ is negative definite, it has a maxima. If $H$ is indefinite, it has a saddle point. Otherwise the test is inconclusive.
My question is what if $f$ is differentiable but non-analytic at $\vec x_o$? In that case, clearly, we can't write it's Taylor expansion, then how do we test? Or am I misunderstanding something?
To find the hermittian of $f,$ you need not write the series expansion. As long as all the required partial derivatives exist, you can always get $H.$