True or false statement?

Question

If $f'(x)!=0$ and $f$ is strictly monotone in an interval $[a,b]subeRR$ then $f(x)>0$ or $f(x)<0$

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Answer 1 · 2018-03-15T23:47:29

False

Consider a counterexample:

Let $f(x)=x(x-1)$ on $[3/4, 7/4]$ . Indeed, we have $f'(x)!=0$ and $f$ monotone on this interval, but we observe that $f(1)=0$ .

So the statement is false.