Real function/Extrema/Higher derivatives/Fact

Let ${}I$ denote a real interval,

$f\colon I\longrightarrow \mathbb {R}$

an ${}(n+1)$-times continuously differentiable function, and ${}a\in I$ an inner point of the interval. Suppose that

$f'(a)=f^{\prime \prime }(a)=\ldots =f^{(n)}(a)=0{\text{ and }}f^{(n+1)}(a)\neq 0$

is fulfilled. Then the following statements hold.

1. If ${}n$ is even, then ${}f$ does not have a local extremum in ${}a$.
2. Suppose that ${}n$ is odd. In case ${}f^{(n+1)}(a)>0$, the function ${}f$ has an isolated local minimum in ${}a$.
3. Suppose that ${}n$ is odd. In case ${}f^{(n+1)}(a)<0$, the function ${}f$ has an isolated local maximum in ${}a$.

Proof, Write another proof