if $f(x)=x^4$ we have critical points given by $$4x^3=0$$ which is $x=0$
Now $$f''(x)=12x^2$$ so $$f''(0)=0$$ and also $x=0$ is not Point of Inflexion since $f''(0^{+})$ and $f''(0^{-})$ have same sign. Now if we take triple derivative $$f'''(x)=24x$$ and $$f''''(0) \gt 0$$
Can we say $x=0$ is Local Minima because from graph of $x^4$,Minima occurs at $x=0$
Yes, there is a general form of the derivative test. Roughly it is as follows.
Let $c$ be a point such that the first $n$ derivatives are zero and the $n+1$th derivative is nonzero. If $n$ is odd, then
if $f^{(n+1)}(c) < 0 $, then $c$ is a local maximum
if $f^{(n+1)}(c) > 0 $, then $c$ is a local minimum
See this link for a more precise and complete answer.
