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Question
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Options
increasing
decreasing
constant
none of these
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Solution
\[f(x) = \frac{- x}{2} + \sin x\text { defined on } \left[ \frac{- \pi}{3}, \frac{\pi}{3} \right]\]
\[ \therefore f'(x) = \frac{- 1}{2} + \cos x \]
\[ \Rightarrow f'(x) \geqslant 0 \forall x \in \left[ \frac{- \pi}{3}, \frac{\pi}{3} \right]\]
\[\left[ \because \text { for } x \in \left[ \frac{- \pi}{3}, \frac{\pi}{3} \right] , \cos x \geqslant \frac{1}{2} \right]\]
Hence, the given function is increasing .
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