Can the derivative $f^\prime$ of a smooth function $f\in C^\infty[0,1]$ change sign infinitely many times (or $f$ have infinitely many isolated critical points)? If yes, how about an analytic function over a closed finite interval or more generally—what do I need to assume about $f$ in order to exclude this possibility? If no, can I get away with $f\in C^k[0,1]$ for some finite $k$. Also, out of curiosity: do you know of a term for functions whose image never change sign? 



Derivative of smooth function change sign infinitely on [0,1]?
Yes, for example $f(x)=e^{-1/x}\sin(1/x)$. No for analytic functions -- such function has all derivatives equal to zero at accumulating point of its zeroes.


