If the function is indeed monotonic, what then is the intuitive meaning of $0$ rate of change of $f(x)$ wrt $x$ at $x=a$?
2026-04-02 07:55:57.1775116557
If the derivative of a function is positive (or negative) at all points except at $x=a$, where it is zero, then monotoncity of function at $x=a$.
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The value of $f'(a)$ is a good predictor of the monoticity behavior of $f$ at $x=a$. If $f'(a) > 0$, then $f$ is increasing, if $f'(a) < 0$ then $f$ is decreasing. The only 'problem case' is $f'(a) = 0$, where that information is not enough to determine the monocity of $f$ at $x=a$. It could be increasing, decreasing, both (if $f$ is a constant) or neither ($x=a$ is a local extremum of $f$).
In that 'problem case', one needs to consult the higher order derivatives to find out how the function behaves locally. If you know about the Tayler series, you can see that the first non-zero derivative determines the behavior. With your description, it seems that the second derivative is also 0 and maybe the third is positive, showing an increasing function.