Question

The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
(a) increasing
(b) decreasing
(c) constant
(d) none of these

Answer 1

$$\begin{gathered} f\left(x\right)=\frac{-x}{2}+\sin x\mathrm{defined}\mathrm{on}\left[\frac{-\mathrm{\pi }}{3},\frac{\mathrm{\pi }}{3}\right] \\ \therefore f\text{'}\left(x\right)=\frac{-1}{2}+\cos x \\ \Rightarrow f\text{'}\left(x\right)⩾0\forall x\in \left[\frac{-\mathrm{\pi }}{3},\frac{\mathrm{\pi }}{3}\right] \\ \left[\because \mathrm{for}x\in \left[\frac{-\mathrm{\pi }}{3},\frac{\mathrm{\pi }}{3}\right],\cos x⩾\frac{1}{2}\right] \end{gathered}$$

Hence, the given function is increasing .