For which values of x is the function f(x) = | vedclass

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(C) Given $f(x) = \sin x + \cos 2x$. First,find the derivative $f'(x) = \cos x - 2 \sin 2x$. Using $\sin 2x = 2 \sin x \cos x$,we get $f'(x) = \cos x

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$\frac{(2n+1)\pi}{2}$

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(C) Given $f(x) = \sin x + \cos 2x$. First,find the derivative $f'(x) = \cos x - 2 \sin 2x$. Using $\sin 2x = 2 \sin x \cos x$,we get $f'(x) = \cos x - 4 \sin x \cos x = \cos x (1 - 4 \sin x)$. For critical points,set $f'(x) = 0$,which gives $\cos x = 0$ or $\sin x = \frac{1}{4}$. Now,find the second derivative $f''(x) = -\sin x - 4 \cos 2x$. For $x = (2n+1)\frac{\pi}{2}$,$\sin x = (-1)^n$ and $\cos 2x = \cos((2n+1)\pi) = -1$. Thus,$f''((2n+1)\frac{\pi}{2}) = -(-1)^n - 4(-1) = 4 - (-1)^n$. If $n$ is even,$f'' = 4 - 1 = 3 > 0$ (Local Minimum). If $n$ is odd,$f'' = 4 - (-1) = 5 > 0$ (Local Minimum). Thus,the function is minimum at $x = (2n+1)\frac{\pi}{2}$.

For which values of $x$ is the function $f(x) = \sin x + \cos 2x$ $(x > 0)$ minimum?

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