Determine whether
Question:

Determine whether $f(x)=x / 2+\sin x$ is increasing or decreasing on $(-\pi / 3, \pi / 3) ?$

Solution:

we have,

$f(x)=-\frac{x}{2}+\sin x$

$=\mathrm{f}^{\prime}(\mathrm{x})=-\frac{1}{2}+\cos \mathrm{x}$

Now,

$x \in\left(-\frac{\pi}{3}, \frac{\pi}{3}\right)$

$\Rightarrow-\frac{\pi}{3}<x<\frac{\pi}{3}$

$\Rightarrow \cos \left(-\frac{\pi}{3}\right)<\cos x<\cos \frac{\pi}{3}$

$\Rightarrow \cos \left(\frac{\pi}{3}\right)<\cos x<\cos \frac{\pi}{3}$

$\Rightarrow \frac{1}{2}<\cos x<\frac{1}{2}$

$\Rightarrow-\frac{1}{2}+\cos x>0$

$\Rightarrow f^{\prime}(x)>0$

Hence, $f(x)$ is an increasing function on $(-\pi / 3, \pi / 3)$

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