The number of points of local maxima of the f | vedclass

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(A) Given the function $f(x) = x + \sin x$. To find the local maxima,we first find the derivative $f'(x) = 1 + \cos x$. Setting $f'(x) = 0$,we get $1

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$0$

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(A) Given the function $f(x) = x + \sin x$. To find the local maxima,we first find the derivative $f'(x) = 1 + \cos x$. Setting $f'(x) = 0$,we get $1 + \cos x = 0$,which implies $\cos x = -1$. This occurs at $x = (2n + 1)\pi$ for any integer $n$. Now,we check the sign of $f'(x)$ around these critical points. Since $\cos x \geq -1$ for all $x$,$f'(x) = 1 + \cos x \geq 0$ for all $x$. Because $f'(x)$ does not change sign from positive to negative at any point $x = (2n + 1)\pi$,the function is strictly increasing and does not have any local maxima. Therefore,the number of points of local maxima is $0$.

The number of points of local maxima of the function $f(x) = x + \sin x$ is-

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