show that the function fx = -x/2 + sinx is strictly increasing on (-pi/3,pi/3)?

Dear student,
$$\begin{gathered} f\left(x\right) = -\frac{x}{2}+\sin x \\ {f}^{\text{'}}\left(x\right) = -\frac{1}{2}+\cos x \\ for strictly increa\sin g f^{\text{'}}\left(x\right)>0 for all x \\ then \\ \cos x>\frac{1}{2} ....1 \\ eq1 is true for x\in (-\frac{\pi }{3},\frac{\pi }{3}) \\ proved f\left(x\right) = -\frac{x}{2}+\sin x is strictly increa\sin g for x\in (-\frac{\pi }{3},\frac{\pi }{3}) \end{gathered}$$

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