What we can say about the monotonocity of the function $f$ near zero

Question

Let us consider a real analytic function $f$. Let us consider the Taylor expansion for $f$ at $x=0$: $$f(x)=b+ax+O(x²)$$

where $b=f(0)$ and $a=f′(0)$

Assuming that $a>0$ so the Taylor expansion for $f$ at $x=0$ is increasing locally near zero.

My question is:

What we can say about the monotonocity of the function $f$ near zero, i.e., Is the function $f$ increasing just like its approximation.

Answer 1

Since $f$ is analytic, $f'$ is continuous. And so, since $f'(0)>0$, you have $f'(x)>0$ on some interval $(-\varepsilon,\varepsilon)$ and therefore, yes, $f$ is strictly increasing on that interval.