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All of the following must be true for a function $f(x)$ that is continuous on the interval $ \left[ a, \, b \right]$ EXCEPT

If $f(a) \le N \le f(b)$, then $f(c) = N$ for at least one $c$ in the open interval $(a, \, b)$.

$f'(c) = \cfrac {f(b) - f(a)}{b-a} $ for at least one $c$ in the open interval $(a, \, b)$.

$f(c)$ is defined for all $c$ in the open interval $(a, \, b)$.

There exists a $c$ in the closed interval $\left[ a, \, b \right]$ such that $f(c) \ge f(x)$ for all other values of $x$ in $\left[ a, \, b \right]$.