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If f(x) has the second order derivative

Question:

If $f(x)$ has the second order derivative at $x=c$ such that $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)>0$, then $c$ is a point of____________

Solution:

Second derivative test: Let $f(x)$ be a function defined on an interval I and $c \in I .$ Suppose $f(x)$ be twice differentiable at $x=c .$ Then, $x=c$ is a point of local minima if $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)>0$. In this case, $f(c)$ is then the local minimum value of $f(x)$.

So, if $f(x)$ has the second order derivative at $x=c$ such that $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)>0$, then $c$ is a point of local minima.

If $f(x)$ has the second order derivative at $x=c$ such that $f^{\prime}(c)=0$ and $f^{\prime \prime}(c)>0$, then $c$ is a point of __local minima___.

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