I have a conjecture on local max/min , can any of you propose a contradiction? If $f$ is a non-piecewise function defined continuous on an interval $I$, and within that interval $I$, there exists a value $x$, such that $f`(x)$ (derivative of $f$) does not exist , then at that value $x$, is a local $\max/\min$ value.

Very not true. For example:
$$f(x) = \begin{cases}x&\text{ if } x\leq 0\\ 2x&\text{ if } x\geq0\end{cases}$$ on $I=[-1,1]$.
