Let $f\colon[a,b]\to\mathbb{R}$ be a differentiable function and $x\in[a,b]$ with $f'(x)=0$. Is there a counterexample with respect to the equivalence
$x$ is not a local extremum of $f$
$\Leftrightarrow$
There is an $\epsilon>0$ such that $f'$ is monotonically increasing on $[x-\epsilon;x]$ and monotonically decreasing on $[x;x+\epsilon]$ or
there is an $\epsilon>0$ such that $f'$ is monotonically decreasing on $[x-\epsilon;x]$ and monotonically increasing on $[x;x+\epsilon]$.
In other words: Are the two definitions of a saddle point
$x$ is a saddle point of a differentiable function $f$ iff $x$ is a stationary point which is not an local extremum.
and
$x$ is a saddle point of a differentiable function $f$ iff $x$ is a stationary inflection point
really equivalent?