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A function $f$ is continuous over $[a,b]$ and differentiable over $(a,b)$.

Which of the following statements does NOT follow from the Mean Value Theorem?

A

If $f\left( b \right) >f\left( a \right) $, then $f^{ \prime }\left( c \right) >0$ for some $c$, where $c$ is contained in $(a,b)$.

B

If $f\left( b \right) =f\left( a \right) $, then $f^{ \prime }\left( c \right) =0$ for some $c$, where $c$ is contained in $(a,b)$.

C

If, for some $c$ contained in $(a,b)$, $f^{ \prime }\left( c \right) =0$, then $f\left( b \right) =f\left( a \right) $.

D

If $f\left( b \right)$ $<$ $f\left( a \right) $, then $f^{ \prime }\left( c \right)$ $ <$ $0$ for some $c$, where $c$ is contained in $(a,b)$.

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