# True or false statement?

## If $f ' \left(x\right) \ne 0$ and $f$ is strictly monotone in an interval $\left[a , b\right] \subseteq \mathbb{R}$ then $f \left(x\right) > 0$ or $f \left(x\right) < 0$

Let $f \left(x\right) = x \left(x - 1\right)$ on $\left[\frac{3}{4} , \frac{7}{4}\right]$. Indeed, we have $f ' \left(x\right) \ne 0$ and $f$ monotone on this interval, but we observe that $f \left(1\right) = 0$.