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AP® Calculus AB-BC

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Mean Value Theorem: Average Rate of Change

APCALC-OAH1BJ

The function $f$ is continuous over the interval $[6,8]$ and differentiable over $(6,8)$.

If the average rate of change of $f$ over $[6,8]$ is $-\frac { 1 }{ 2 } $, which of the following statements must be true?

A

There exists some $c$ on $(6,8)$ such that $f^{ \prime }\left( c \right) =-\cfrac { 1 }{ 2 } $.

B

There exists some $c$ on $(6,8)$ such that $f^{ \prime }\left( c \right) =0 $.

C

$\cfrac { f^{ \prime }\left( 8 \right) -f^{ \prime }\left( 6 \right) }{ 8-6 } =-\cfrac { 1 }{ 2 } $

D

For all $x$ on $(6,8)$, $f^{ \prime }\left( x \right) <0$.