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Examine the applicability of Rolle’s theorem for the function f(x) = \sin 2x in [0, \pi]. Hence find the points where the tangent is parallel to x-axis.

 

 
 
 
 
 

Answers (1)
S safeer

f(x) = \sin 2x

\\$(i) $\mathrm{f}(\mathrm{x})$ is continuous on the closed interval $[0, \pi] \\$ (ii) $\mathrm{f}(\mathrm{x})$ is differentiable in the open interval $(0, \pi)$

\\ $ (iii) \mathrm{f}(0)=\sin ^{2}0 = 0$ $ \\ \mathrm{f}(\pi)=\sin ^{2} \pi=0 \\ $ $\mathrm{f}(0)=\mathrm{f}(\pi)$

The conditions for Rolles theorm is satisfied. 

Now 

\begin{aligned} &\mathrm{f}^{\prime}(\mathrm{x})=2 \sin \mathrm{x} \cos \mathrm{x}\\ &\mathrm{f}^{\prime}(\mathrm{c})=0 \\ & 2 \sin \mathrm{c} \cos \mathrm{c}=0\\ &\sin 2 \mathrm{c}=0\\ &2 \mathrm{c}=0, \pi \ldots\\ &\mathrm{c}=0, \frac{\pi}{2} \ldots\\ &\mathbf{c}=\frac{\pi}{2} \in(0, \pi)\\ \end{aligned}

At x= \frac{\pi}{2} the tangent  is parallel to x-axis.

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