Is $x_0$ local minimum/maximum or inflection point if $f'(x_0)=0$? Is the following statement true or false? (I think it is true, and I tried proving it by contradiction and failed.) 

Let $f:(a,b)\to \mathbb{R}$ be differentiable, and let $x_0\in (a,b)$ such that $f'(x_0)=0$. Then $x_0$ is a local minimum/maximum or inflection point.
This is false. Let $$f(x)=\begin{cases}x^2 \sin (1/x) &amp; x \neq 0 \\ 0 &amp; x=0\end{cases}$$
Then $f$ is differentiable and $f'(0)=0$, but $0$ is neither a local extremum nor an inflection point.
If $f'(x_0) = 0$, then the tangent to the curve $y = f(x)$ at $x = x_0$ is horizontal. Thus it is as you say.