How to determine if derivative changes sign when it's undefined at a point?

Question

I had to find the absolute extrema of $f(x) = 3\sqrt[3]{x^2}-2x$ in the range $[-1;2]$. So first I found that $f'(x) = 2x^{-\frac{1}{3}}-2$. Then I found it to be equal to 0 at the point 1. I then looked at the graph of both $f(x)$ and $f'(x)$ and realized that I missed $f'$ being undefined at 0. Afterwards I found the correct answers, assuming that $f'$ changes sign at the point 0, but how am I supposed to actually check if that's the case? Thinking about it, why can't the derivative be undefined at a point, but have the same sign both to the left and right of that point?

Answer 1

While phrased in terms of absolute extrema, this is really more about relative extrema (relative can end up being absolute via further checking).

In many examples, you find $x$-values where $f'(x)=0$ as candidates for relative extrema, then determine if they actually are by either the second derivative test or graphing. As you likely know, such values can be neither relative maxima nor minima - e.g. $f(x)=x^3$ has an inflection point when $x=0$ even though $f'(0)=0$.

Relative extrema occur where $f'(x)$ changes sign, which includes points where it does not exist, not just equal to $0$. In other words, you need to check these also, which is your situation above - $f'(0)$ does not exist, but it is a relative minimum and graphically it's not a "nice" low point like in a parabola, but rather a cusp.

To your last question - it CAN happen that the derivative has the same sign on both sides of a point where it does not exist - take a look at $f(x) = \sqrt[3]{x}$.