I understand that the values of $x$ that allow $f'(x)=0$ are stationary points and therefore potential local maximums and minimums of $f(x)$. When would a stationary point NOT be a local maximum or minimum? Do inflection points also yield $f'(x)=0$?
not always, for example $f(x)=x^3+x$ has an inflection point in x=0 but $f’(0)=1$
to be more precise, when derivatives exist:
if $f’(x_0)=0$ and $\exists k \geq 2$ s.t. $f^k(x_0) \neq 0$ then
when k is even you have a max/min in $x_0$ (depending on the sign)
when k is odd you have an inflection point in $x_0$
[EDIT] I've added up my 2 answers and corrected a typo
If x is an inflection point for f then the second derivative, $f″(x)$ is equal to zero if it exists.
Thus, no, inflection points do not give $f'(x)=0$ necessarily, but $f''(x)=0$.
Regarding stationary points and $f'(x)=0$ and stationary points, consider the function :
$$f(x) = x^3$$
Then, it is :
Which is equal to zero at $x=0$, but this is not a stationary point of the function.