Determine Whether the Mean Value Theorem Can Be Applied to the Function

Multiple Choice

Determine whether the Mean Value Theorem can be applied to the function $f ( x ) = 2 \sin x + \sin 2 x$ on the closed interval $[ 4 \pi , 5 \pi ]$ . If the Mean Value Theorem can be applied, find all numbers $c$ in the open interval $( 4 \pi , 5 \pi )$ such that $f ^ {t } ( c ) = \frac { f ( 5 \pi ) - f ( 4 \pi ) } { 5 \pi - 4 \pi }$ .

A) MVT applies; $\frac { 9 \pi } { 2 }$
B) MVT applies; $\frac { 13 \pi } { 4 }$
C) MVT applies; $\frac { 13 \pi } { 6 }$
D) MVT applies; $\frac { 13 \pi } { 3 }$
E) MVT does not apply