Maxima or minima in a function

Question

Suppose if a function is given which is differentiable throughout it's domain.This function has a line of symmetry say x=a, passing through the function.Then is it always true that the point at which this line intersects this function ,say(a,b) is a point of minima or maxima?

Answer 1

It's true that $f'(a)=0$ necessarily, but $a$ may be neither a local minimum nor a local maximum. For instance, $$f(x)=\begin{cases}e^{-x^{-2}}\sin(x^{-2})&\text{if }x\ne 0\\ 0&\text{if }x=0\end{cases}$$ is $C^\infty$ and even (i.e. the case $a=0$), but it is frequently positive and negative in all neighbourhoods of $0$.

Answer 2

If I understood the question correctly, then a counter-example would be $$ f(x)=|x|\sin(1/|x|) $$ whose graph is: