\[f(x)\]
is monotonic (increasing) if \[\frac{d(f(x))}{dx} \ge 0\]
or monotonic (decreasing) if \[\frac{d(f(x))}{dx} \le 0\]
on its domain.It is strictly monotonic (increasing) if
\[\frac{d(f(x))}{dx} \gt 0\]
and strictly monotonic (decreasing) if \[\frac{d(f(x))}{dx} \lt 0\]
on its domain.It is not enough that
\[\frac{d(f(x))}{dx} \neq 0\]
for a function to be strictly monotonic. \[f(x)= \frac{1}{x}\]
is strictly monotonic (decreasing) for \[x \lt 0\]
and strictly monotonic (increasing) for \[x \gt 0\]
but at \[x=0\]
\[f(x)\]
is not defined.A monotonic function has no turning points, though it can have stationary points of inflection (
\[\frac{d(f(x))}{dx}= \frac{d^2 (f(x))}{dx^2}=0\]
). If it is strictly monotonic, it has no stationary points so that \[\frac{d(f(x))}{dx} \neq 0\]
.