It is known that in order to find the local extrema of a function, one must find its critical points (where the derivative is zero or undefined), followed by which one would substitute these values into the function's second derivative and if it's positive, it's a local minimum, and if it's negative, it's a local maximum.
But what if the second derivative is equal to zero? How are the local extrema evaluated then? Must we take the third derivative, or is it something simpler I'm missing?
A further sub-question is about how to determine if a point on a function is a local extremum if all derivatives at that point are undefined. For example, the one local (and absolute) minimum of $f(x) = |x|$ is obviously $(0,0)$, but how would one prove this algebraically, considering that $f^{(n)}(x)$ does not exist?