If a function is strictly monotonic increasing after the derivative test for $(-\infty, 0)$ and $(0, \infty)$, does that imply that it's strictly monotonic increasing for $(-\infty, \infty)$?
As an example, $f(x) = x^3$, $f'(x) = 3x^2$, so it's increasing for all $x \in \mathbb{R}$, $x \neq 0$ after the derivative test, but it's actually for all $x \in \mathbb{R}$