Is it possible to find $\eta>0$ such that $\forall x\in (\alpha-\eta,\alpha+\eta),\qquad \dfrac{f''(x)}{f'(x)}<1$

Question

Let $I$ an open interval and $f\in\mathcal{C}^2(I)$, assume that $\alpha\in I$ such that $f(\alpha)=0$ and $f'(\alpha)\ne0$

Is it possible to find $\eta>0$ such that $\forall x\in (\alpha-\eta,\alpha+\eta),\qquad \dfrac{f''(x)}{2f'(x)}<1$

Answer 1

No : if you take for exemple $f(x)=\exp(x)-\exp(\alpha)$ for your title, and $f(x)=\exp(2x)-\exp(2\alpha)$ for your inside text. You will have for all $x\in\mathbb{R}$, respectively $f''(x)/f'(x)=1$ and $f''(x)/(2f'(x))=1$.