Real function/Extrema/Second derivative/Fact/Proof/Exercise

Let ${}I$ be a real interval,

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

a twice continuously differentiable function, and let ${}a\in I$ be an inner point of the interval. Suppose that ${}f'(a)=0$. Show the following statements.

1. If ${}f^{\prime \prime }(a)>0$ holds, then ${}f$ has an isolated local minimum in ${}a$.
2. If ${}f^{\prime \prime }(a)<0$ holds, then ${}f$ has an isolated local maximum in ${}a$.

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