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Question
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
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Solution
\[ f\left( x \right) = \sin x + \left| \sin x \right|, 0 < x \leq 2\pi\]
\[\text { Case I: When x }\in \left( 0, \pi \right)\]
\[f\left( x \right) = \sin x + \sin x = 2\sin x\]
\[ \Rightarrow f'\left( x \right) = 2\cos x\]
\[\text { As,} \cos x > 0 \text { for } x \in \left( 0, \frac{\pi}{2} \right) \text { and }\cos x < 0 \text { for } x \in \left( \frac{\pi}{2}, \pi \right)\]
\[\text { So,} f'\left( x \right) > 0\text { for} x \in \left( 0, \frac{\pi}{2} \right)\text{ and } f'\left( x \right) < 0 \text { for }x \in \left( \frac{\pi}{2}, \pi \right)\]
\[ \therefore f\left( x \right)\text { is increaing on} \left( 0, \frac{\pi}{2} \right) \text { and } f\left( x \right) \text { is decreasing on } \left( \frac{\pi}{2}, \pi \right) . \]
\[\text { Case II: When x } \in \left( \pi, 2\pi \right)\]
\[f\left( x \right) = \sin x - \sin x = 0\]
\[ \Rightarrow f'\left( x \right) = 0\]
\[\text { So,} f\left( x \right) \text { is neither increaing nor decreasing on } \left( \pi, 2\pi \right) . \]
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