extrema of function In my class we learned the following theorem: 
If $f$ is an $n$-times differentiable function at a point $x$ and all the derivatives up until $m$ are equal to $0$ with $m$ being different than $0$ then we know:
If $m$ is even than $x$ is an extrema (min or max depending on the sign of $f(x)$).
and if $m$ is odd then $x$ is an inflection point (ascending or descending again depending on the sign).
Could you give me a hint on why when $m$ is even we know $x$ is an extrema and when its odd we know we have an inflection point?

hint
By Taylor-Young, we will have, for $t$ near $x$,
$$f(t)-f(x)=$$
$$\frac{(t-x)^m}{m!}f^{(m)}(x)\Bigl(1+\epsilon(t)\Bigr)$$
with $$\lim_{t\to x}\epsilon(t)=0.$$
So, if $m$ is even, the sign of  $f(t)-f(x)$
do not change. It is an extrema.
