- $f(x)$ is a convex function and twice differentiable. The domain is $\mathbb{R}$.
- suppose $f'(x_{0})=0$ , and $x_{0}$ is unique.
- My question is: what is $inf(\frac{x-x_{0}}{f'(x)})$,$x\neq x_{0}$?
I'm aware that this function has a limit when x approaching $x_{0}$ and $f''(x_{0})$ is not $0$ (despite that the limit may not be important in this question).
- $\lim_{x\rightarrow x_{0}}\frac{x-x_{0}}{f'(x)}=\frac{1}{f''(x_{0})}$
But can we always get a infimum of this set? I tried Taylor decomposition， apparently the sign of the Lagrange residue is uncertain (even it is third order differentiable).
- $f'(x_{0})=f'(x)+f''(x)(x_{0}-x)+\frac{f'''(\varepsilon )}{2}(x_{0}-x)^2$
the question is can we always get a infimum of $\frac{x-x_{0}}{f'(x)}$? if yes, what is it? if not, what condition can make this true.