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

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Rolle's Theorem states that if a function $f$ is continuous on the 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 $f'(c) = 0$. For $f(x) = \sin(5\pi x)$ on $\left[-\frac{2}{5}, \frac{2}{5}\right]$, the function is continuous and differentiable everywhere, and $f\left(-\frac{2}{5}\right) = f\left(\frac{2}{5}\right) = 0$, satisfying all three hypotheses of Rolle's Theorem. To find $c$, we differentiate $f(x)$ to get $f'(x) = 5\pi \cos(5\pi x)$ and set it equal to zero to find the values of $x$ where the slope of the tangent is zero. Solving $5\pi \cos(5\pi x) = 0$ gives $x = \pm \frac{1}{10}, \pm \frac{3}{10}$, which are the points within the interval where the function's derivative equals zero, satisfying Rolle's Theorem.

Verify That the Function Satisfies the Three Hypotheses of Rolle's f(x)=sin⁡5πx,[−25,25]f ( x ) = \sin 5 \pi x , \left[ - \frac { 2 } { 5 } , \frac { 2 } { 5 } \right]f(x)=sin5πx,[−52​,52​]

Verify That the Function Satisfies the Three Hypotheses of Rolle's $f ( x ) = \sin 5 \pi x , \left[ - \frac { 2 } { 5 } , \frac { 2 } { 5 } \right]$