Which function is differentiable on $[a,b]$ but doesn't reach an extreme on $[a,b]$?

Question

I think this is an old question but I can't find any evidence. When I learn about Flett's theorem and Darboux's theorem, we consider it to have an extrema on $(a,b)$ and on two boundaries. But in case it does not reach the extreme at $(a,b)$, it must reach the boundary. As far as I know, when a function is differentiable on $[a,b]$, its graph must always be 'smooth', meaning no sharp corners. Obviously then the peak of the smooth line must be the extreme value. Does this mean that function $f$ is an increasing function?

Answer 1

Let's consider $$f(x)=\begin{cases}x^4\left(2+\sin\frac{1}{x} \right), & x\neq 0\\ 0, & x=0 \end{cases}$$ function has absolute minimum in $x=0$, but its derivative $$f'(x)=\begin{cases}x^2\left[4x\left(2+\sin\frac{1}{x} \right)-\cos\frac{1}{x}\right], & x\neq 0\\ 0, & x=0 \end{cases}$$ doesn't keep sign in any one-sided neighbourhood of zero. Hope, this answers as body of question, so comments.

Additionally let me bring function which has positive derivative in point, but is not monotone in any neighbourhood of point: $$f(x)=\begin{cases}x+2x^2\sin\frac{1}{x} , & x\neq 0\\ 0, & x=0 \end{cases}$$ with derivative $$f'(x)=\begin{cases}1+4x\sin\frac{1}{x}-2 \cos\frac{1}{x}, & x\neq 0\\ 1, & x=0 \end{cases}$$