Hessian under integral sign

Question

If I have a function $f(a,b)$ s.t. $f(a,b)=\int_{I}{f(x,a,b)dx}$, how would I be able to characterize the point $(a^{*},b^{*})$ where $f(a^{*},b^{*})$ is an extremum? Would the Hessian simply be $$\begin{vmatrix}\int_{I}f_{aa}(x,a,b)dx&\int_{I}f_{ab}(x,a,b)dx\\\int_{I}f_{ba}(x,a,b)dx&\int_{I}f_{bb}(x,a,b)dx\end{vmatrix}$$

Answer 1

If $P=(a_0,b_0)$ is a critical point, i.e. solution of both $f_a=0$ and $f_b=0,$ AND the determinant of the Hessian at $P$ (what you have displayed, but with coordinates of critical point plugged in) is positive, then that tells you there is either a local max or a local min at $P.$ You have to look say at the first partials near $P$ to tell which. On the other hand if the Hessian determinant is negative there is a saddle point at $P$ so neither local max nor local min there. Finally a zero Hessian determinant gives no information, as one sees by a few examples.

Keep in mind all this is predicated on smoothness of $f(a,b)$ near $P,$ and that only the local max/min is decided, not the global or absolute max over whatever region $(a,b)$ ranges over.