Question

Determine whether $f\left(x\right)=-\frac{x}{2}+\sin x$ is increasing or decreasing on $(-\frac{\pi }{3},\frac{\pi }{3})$.

Answer 1

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

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

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

$\frac{\sqrt{3}}{2}<\cos x<\frac{\sqrt{3}}{2}$

$\frac{\sqrt{3}}{2}<\cos x<1$

$\frac{\sqrt{3}-1}{2}<\cos x-\frac{1}{2}<\frac{1}{2}$

$f^{\prime }\left(x\right)>0$ & $x\in (\frac{-\pi }{3},\frac{\pi }{3})$

$\therefore f\left(x\right)$ is increasing on $(\frac{-\pi }{3},\frac{\pi }{3})$