The value c of the Rolle's theorem for the fu | vedclass

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(D) Given function $f(x) = 2 \sin x + \sin 2x$ on $[0, \pi]$.Since $f(x)$ is a sum of trigonometric functions,it is continuous on $[0, \pi]$ and diffe

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$\frac{\pi}{3}$

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(D) Given function $f(x) = 2 \sin x + \sin 2x$ on $[0, \pi]$.Since $f(x)$ is a sum of trigonometric functions,it is continuous on $[0, \pi]$ and differentiable on $(0, \pi)$.Also,$f(0) = 2 \sin(0) + \sin(0) = 0$ and $f(\pi) = 2 \sin(\pi) + \sin(2\pi) = 0$.Thus,$f(0) = f(\pi)$,satisfying the conditions of Rolle's theorem.We find $f'(x) = 2 \cos x + 2 \cos 2x$.Setting $f'(c) = 0$,we get $2 \cos c + 2 \cos 2c = 0$,which implies $\cos c + \cos 2c = 0$.Using the identity $\cos 2c = 2 \cos^2 c - 1$,we have $2 \cos^2 c + \cos c - 1 = 0$.Factoring the quadratic equation: $(2 \cos c - 1)(\cos c + 1) = 0$.This gives $\cos c = \frac{1}{2}$ or $\cos c = -1$.For $c \in (0, \pi)$,$\cos c = \frac{1}{2}$ implies $c = \frac{\pi}{3}$.Since $\cos c = -1$ gives $c = \pi$,which is not in the open interval $(0, \pi)$,the only valid value is $c = \frac{\pi}{3}$.

The value $c$ of the Rolle's theorem for the function $f(x) = 2 \sin x + \sin 2x$ in the interval $[0, \pi]$ is

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