Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers $c$ that satisfy the conclusion of Rolle's Theorem.
$$f ( x ) = \sin 5 \pi x , \left[ - \frac { 2 } { 5 } , \frac { 2 } { 5 } \right]$$
A) $c _ { 1 } = \pm \frac { 1 } { 10 } , c _ { 2 } = \pm \frac { 3 } { 10 }$
B) $c _ { 1 } = \pm \frac { 1 } { 10 } , c _ { 2 } = \frac { 3 } { 10 }$
C) $c _ { 1 } = - \frac { 1 } { 10 } , c _ { 2 } = \pm \frac { 1 } { 10 }$
D) $c _ { 1 } = \frac { 1 } { 10 } , c _ { 2 } = \pm \frac { 3 } { 10 }$
E) None of these

Question

Quizplus · Accepted Answer

The function $f(x) = \sin 5\pi x$ is continuous and differentiable on the interval $[- \frac{2}{5}, \frac{2}{5}]$. Also, $f(-\frac{2}{5}) = f(\frac{2}{5}) = 0$, satisfying the conditions of Rolle's Theorem. The derivative is $f'(x) = 5\pi \cos 5\pi x$. Setting $f'(x) = 0$, we find $x = \pm \frac{1}{10}, \pm \frac{3}{10}$ within the interval.

Verify That the Function Satisfies the Three Hypotheses of Rolle's ccc

Verify That the Function Satisfies the Three Hypotheses of Rolle's $c$