The Number of values of $C$ that satisfy the conclusion of Rolle's theorem in case of following function $\mathrm{f}(x)=\sin 2 \pi x, x \in[-1,1]$ is

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Aicharya · Accepted Answer

$\begin{array}{ll} & \mathrm{f}(x)=\sin 2 \pi x \\ \therefore \quad & \mathrm{f}^{\prime}(x)=2 \pi \cos 2 \pi x \\ & \text { Now, } \mathrm{f}^{\prime}(\mathrm{C})=0 \\ \Rightarrow & 2 \pi \cos 2 \pi \mathrm{C}=0\end{array}$ $\begin{aligned} & \Rightarrow \cos 2 \pi \mathrm{C}=0 \\ & \Rightarrow 2 \pi \mathrm{C}=\frac{-3 \pi}{2}, \frac{-\pi}{2}, \frac{\pi}{2}, \frac{3 \pi}{2} \\ & \quad \ldots[\because x \in(-1,1) \Rightarrow 2 \pi x \in(-2 \pi, 2 \pi)] \\ & \Rightarrow \mathrm{C}=\frac{-3}{4},-\frac{1}{4}, \frac{1}{4}, \frac{3}{4} \end{aligned}$
Number of values of $\mathrm{C}=4$

The Number of values of $C$ that satisfy the conclusion of Rolle's theorem in case of following function…

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