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AP® Calculus AB-BC

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Moderate

Mean Value Theorem: When It Fails

APCALC-Y@OC1E

A function $g$ is continuous for $-3\le x\le 3$. Additionally, $g(-3)=12$ and $g(3)=24$.

If there exists no value of $c$ over $-3 < c < 3$ for which $g^{ \prime }\left( c \right) =2$, which statement below must be true?

A

$g^{ \prime }\left( k \right) <0$ for all $k$, where $-3<$ $k$ $<3$.

B

$g^{ \prime }\left( k \right)$ does not exist for some $k$, where $-3<$ $k$ $<3$.

C

$g^{ \prime }\left( k \right)=0$ for some $k$, where $-3<$ $k$ $<3$.

D

$g\left( k \right) =0$ for some $k$, where $-3<$ $k$ $<3$.