At what value of x does the function f(x) = s | vedclass

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(C) Let $f(x) = \sin x(1 + \cos x) = \sin x + \sin x \cos x = \sin x + \frac{1}{2} \sin 2x$.To find the critical points,we differentiate $f(x)$ with r

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$x = \pi / 3$

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(C) Let $f(x) = \sin x(1 + \cos x) = \sin x + \sin x \cos x = \sin x + \frac{1}{2} \sin 2x$.To find the critical points,we differentiate $f(x)$ with respect to $x$:$f'(x) = \cos x + \cos 2x$.Set $f'(x) = 0$ for extrema:$\cos x + \cos 2x = 0 \implies \cos 2x = -\cos x$.Using the identity $-\cos x = \cos(\pi - x)$:$\cos 2x = \cos(\pi - x) \implies 2x = \pi - x \implies 3x = \pi \implies x = \pi / 3$.Now,find the second derivative $f''(x)$ to check for maxima:$f''(x) = -\sin x - 2 \sin 2x$.At $x = \pi / 3$:$f''(\pi / 3) = -\sin(\pi / 3) - 2 \sin(2\pi / 3) = -\frac{\sqrt{3}}{2} - 2(\frac{\sqrt{3}}{2}) = -\frac{\sqrt{3}}{2} - \sqrt{3} = -\frac{3\sqrt{3}}{2}$.Since $f''(\pi / 3) < 0$,the function has a maximum value at $x = \pi / 3$.

At what value of $x$ does the function $f(x) = \sin x(1 + \cos x)$ have a maximum value?

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