Does $f(x)$ has inflection point at $c$ if $f(x)$ is diffrentiable at interval $I$ and $f'(x) > 0$ at $I$ except $c$? Does $f(x)$ has inflection point at $c$ if $f(x)$ is differentiable at interval $I$ and $f'(x) > 0$ at $I$ except at $c$ (interval $I$ includes $c$)?
I think it is geometrically true, but I can't prove strictly.  

By Darboux theorem $f'$ has IVP. If $f'(c) <0$ then any number between $f'(c)$ and $0$ will  not be attained by $f'$ contradicting IVP. Hence it follows that $f'(c)$ must be $0$. 
