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} $\Rightarrow \cos \left(-\frac{\pi}{3}\right)<\cos x<\cos \frac{\pi}{3}$ $\Rightarrow \cos \left(\frac{\pi}{3}\right)<\cos x<\cos \frac{\pi}{3}$ $\Rightarrow \frac{1}{2}<\cos x<\frac{1}{2}$ $\Rightarrow-\frac{1}{2}+\cos x>0$ $\Rightarrow f^{\prime}(x)>0$ Hence, $f(x)$ is an increasing function on $(-\pi / 3, \pi / 3)$
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