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Question
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
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Solution
\[f\left( x \right) = \sin x - \cos x, x \in \left( 0, 2\pi \right)\]
\[f'\left( x \right) = \cos x + \sin x\]
\[\text { For f(x) to be increasin, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \cos x + \sin x > 0\]
\[ \Rightarrow \sin x > - \cos x\]
\[ \Rightarrow \tan x > - 1\]
\[ \Rightarrow x \in \left( 0, \frac{3\pi}{4} \right) \cup \left( \frac{7\pi}{4}, 2\pi \right)\]
\[\text { So,f(x)is increasing on } \left( 0, \frac{3\pi}{4} \right) \cup \left( \frac{7\pi}{4}, 2\pi \right) . \]
\[\text { For f(x) to be decreasing we must have},\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow \cos x + \sin x < 0\]
\[ \Rightarrow \sin x < - \cos x\]
\[ \Rightarrow \tan x < - 1\]
\[ \Rightarrow x \in \left( \frac{3\pi}{4}, \frac{7\pi}{4} \right)\]
\[\text { So,f(x)is decreasing on }\left( \frac{3\pi}{4}, \frac{7\pi}{4} \right).\]
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