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A continuous function $f(x)$ is defined on the set of all real numbers. Consider the closed interval $[a, b]$, with $ a < c < b $.

If $f’ (c) = 0$ and $f'' (x) > 0$ on the interval $[a,\,b]$, then which of the following must be TRUE?

There are local maxima at both $x=a$ and $x =b$.

$f$ is differentiable for all real numbers.

$f(c)$ is a local minimum.

$f(c)$ is a local maximum.