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Question
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
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Solution
\[f\left( x \right) = \frac{- x}{2} + \sin x\]
\[f'\left( x \right) = \frac{- 1}{2} + \cos x\]
\[\text { Here, }\]
\[ \frac{- \pi}{3} < x < \frac{\pi}{3}\]
\[ \Rightarrow \cos x > \frac{1}{2}\]
\[ \Rightarrow \frac{- 1}{2} + \cos x > 0\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in \left( \frac{- \pi}{3}, \frac{\pi}{3} \right)\]
\[\text {So,f }\left( x \right) \text { is increasing on }\left( \frac{- \pi}{3}, \frac{\pi}{3} \right).\]
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