Determine whether or not the function $f ( x ) = \cos 2 x$ satisfies the hypothesis of Rolle's Theorem on the interval $\left[ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } \right]$ If it does, find the exact values of all values of $c \in \left( \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } \right)$ that satisfy the conclusion of Rolle's Theorem.
A) No
B) Yes, $c = \frac { \pi } { 2 } , c = \frac { 3 \pi } { 2 }$
C) Yes, $c = - \frac { \pi } { 2 } , c = \frac { \pi } { 2 }$
D) Yes, $c = \frac { \pi } { 2 }$

Question

Quizplus · Accepted Answer

Rolle's Theorem states that if a function $f$ is continuous on a closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c$ in $(a, b)$ such that $f'(c) = 0$. The function $f(x) = \cos 2x$ is continuous and differentiable everywhere, and $f\left(\frac{\pi}{4}\right) = f\left(\frac{3\pi}{4}\right) = \cos\left(\frac{\pi}{2}\right) = 0$, satisfying the conditions of Rolle's Theorem. The derivative $f'(x) = -2\sin 2x$, and setting $f'(x) = 0$ gives $x = \frac{\pi}{2}$ as the only solution in the interval $\left(\frac{\pi}{4}, \frac{3\pi}{4}\right)$.

Determine Whether or Not the Function f(x)=cos⁡2xf ( x ) = \cos 2 xf(x)=cos2x

Determine Whether or Not the Function $f ( x ) = \cos 2 x$