Determine whether Rolle's Theorem can be applied to the function $f ( x ) = \cos \pi x \text { on }$ the closed interval $\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$. If Rolle's Theorem can be applied, find all numbers $c$ in the open interval $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ such that $f ^ { t } ( c ) = 0$.
A) Rolle's Theorem applies; $c = 0$
B) Rolle's Theorem applies; $c = - \frac { 1 } { 4 } , \frac { 1 } { 4 }$
C) Rolle's Theorem applies; $c = 0 , - \frac { 1 } { 4 } , \frac { 1 } { 4 }$
D) Rolle's Theorem applies; $c = \frac { 1 } { 4 }$
E) Rolle's Theorem does not apply.

Question

Quizplus · Accepted Answer

Rolle's Theorem applies because $f(x) = \cos \pi x$ is continuous on $\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$ and differentiable on $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$, with $f\left(-\frac{1}{2}\right) = f\left(\frac{1}{2}\right)$. The derivative $f'(x) = -\pi \sin \pi x$ equals zero when $x = 0$, which is within the interval.

Determine Whether Rolle's Theorem Can Be Applied to the Function f(x)=cos⁡πx on f ( x ) = \cos \pi x \text { on }f(x)=cosπx on

Determine Whether Rolle's Theorem Can Be Applied to the Function $f ( x ) = \cos \pi x \text { on }$