Using the Second Derivative Test to Determine Extrema AB & BC

If $f'(c) = 0$, then $f''(c) > 0$ makes $f(c)$ a relative minimum and $f''(c) < 0$ makes it a relative maximum: a horizontal tangent inside a bowl is the bottom of the bowl. If $f''(c) = 0$ the test is inconclusive and returns nothing at all. That is not a technicality: $x^{4}$, $-x^{4}$ and $x^{3}$ all have $f'(0) = 0$ and $f''(0) = 0$, and they have a minimum, a maximum, and no extremum respectively, so every input the test reads is identical while the answers exhaust the possibilities.

The test also requires $f'(c)$ to be zero rather than merely critical, so it says nothing at the corner of $|x|$, where the First Derivative Test gives a minimum immediately. Inflection points obey a parallel rule: $f''$ must change sign, and vanishing is neither necessary nor sufficient. $x^{4}$ has $f''(0) = 0$ and no inflection point, $x^{1/3}$ has no $f''(0)$ at all and does have one, and $\frac{1}{x}$ changes concavity across a point that is not in its domain. Report an inflection as a point, with its $y$-coordinate taken from $f$.