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Let $f : [a, b] \rightarrow R$ be differentiable. If $f'(x) \neq 0$ for all $x \in [a, b]$, then show that either $f'(x) \geq 0$ for all $x\in [a, b]$ or $f'(x) \leq 0$ for all $x \in [a, b].$

I'm not sure how I can exactly reach to the result. I have to show $f(x)$ is either strictly monotonic increasing or strictly monotonic decreasing. Please solve.

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    I know Rolle's theorem. please give a hint – 2017-01-15
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    That is "Darboux's Theorem" or "Intermediate value theorem for the derivative", see e.g. http://math.stackexchange.com/questions/270919/intermediate-value-theorem-and-continuity-of-derivative or https://en.wikipedia.org/wiki/Darboux's_theorem_(analysis). – 2017-01-15

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Rolle's theorem implies that this function is injective. Indeed, suppose there exist $x

Since $f$ is differentiable, it is a continuous function.

Then, you conclude by saying that continuous injective function are indeed monotone.