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.

False. Consider, for example,
$$f(x) = \begin{cases} x^3 & \text{ if } x<0 \\ x & \text{ if } x \ge 0\end{cases}.$$
Then $f$ is not differentiable at $0$, but $f$ is strictly increasing around $0$.
