Theorem 5.8 Baby Rudin. Some questions This is with reference to the page number 107 of Baby Rudin. 

Theorem 5.8 Let $f$ be defined on $[a,b]$.If a point $x\in(a,b)$ is a
  local maximum of function $f$, and if $f'(x)$ exist, then $f'(x)=0$.

I have some questions. Thanks in advance for reading and helping out.


*

*What about the converse of this theorem? Suppose $f'(x)=0$ at some point, can we always conclude that it is local maximum? 

*The theorem add a condition that $f'(x)$ should exist. This raised a question, Can we have a function function with local maximum/minimum at some point say $p$ but derivative undefined at the point.
Edits: 
We can take $f(x)=x^2$ on $[-1,1]$. $f'(0)=0$ but $0$  is not local maximum. I think this works for (1).

Answer to question 1
The derivative of $f(x)=x^3$ vanishes at $0$ but $f$ doesn’t have a minimum nor at maximum at $0$.
Answer to question 2
Take $g(x)=\vert x \vert$, again look at zero.
