This question already has an answer here:
To find the critical points of any given function $f$, we should first set the first derivative $f'=0$ and solve to find the values of x.
Now to find at which x=a we have a maxima or a minima, we should find the $nth$ derivative such that $f^n(a)≠0$.
- if n is odd, a is neither a local maximum nor a local minimum.
- if n is even and $f^n(a)>0$, then f has a local minimum at a.
- if n is even and $f(n)(a)<0$, then f has a local maximum at a.
What is the intuition behind these 3 points, in other words why these rules work out.
N.B. - I understand how it works till n=2. But when $f''$ turns out to be zero and we continue to find the next derivatives, that's when I start to feel I don't understand what is actually happening.