Doubts regarding maxima and minima

Question

Does maxima or minima occurs at a point where $f'(x)=\infty$?

If not explain why?

Thanks in advance.

Answer 1

They can, but they neednt. Compare $$f(x) = \sqrt{x}$$ and $$g(x)=\begin{cases} -\sqrt{-x} & \text{for } x\le 0 \\ \sqrt x & \text{for } x\ge 0 \end{cases}$$

Answer 2

If $x$ is a point of the interior of the domain of $f$, and $f$ is derivable, then $x$ is an extrema if, and only if, $f'(x)=0$.

Problems occur when $x$ is on the boundary.

For example, take

$$x\mapsto \sqrt[3]x$$

defined on $[0,1]$.

Then $0$ is a minima for $f$ on $[0,1]$, and "$f'(0)=\infty$".

So yes, it can happens.

Answer 3

Consider $f$ which graph consists of two lower semicircles with radia $1$ and centers $(1,1)$, $(-1,1)$. What happens at $x_0=0$?