What to do *rigorously* when the second derivative test is inconclusive? How do you rigorously check if a point is a local minimum when the second derivative test is inconclusive?
Does there exist a way to do this in general for arbitrary smooth (or analytic...) functions?
I know I can graph the function, plug in a few values, etc. (there are lots of questions with these as answers) but that's not rigorous. I'm wondering what I would do in a rigorous proof -- I don't think I ever learned this.

This answer is for functions of one variable. First, a comment. The second derivative test is often not the best approach. For one thing, it involves computing and evaluating the second derivative. That involves work, and carries a non-zero probability of error. 
In many of the usual calculus problems, the right approach is to look at the behaviour of the first derivative near the critical point. If the first derivative is non-positive in an interval $(a-\epsilon,a)$ of the critical point $a$, and non-negative in an interval $(a,a+\epsilon)$, then we have a local minimum at $a$.
Now assume that $f$ is analytic in an interval about the critical point $a$. Let $k$ be the smallest integer $\ge 2$ such that $f^{(k)}(a)\ne 0$. If $k$ is even, and 
$f^{(k)}(a)\gt 0$, then we have a local minimum at $a$. If $k$ is even, and 
$f^{(k)}(a)\lt 0$, then we have a local minimum at $a$. If $k$ is odd, we have neither a local maximum nor a local minimum at $x=a$. 
