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Question
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
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Solution
\[f\left( x \right) = \sin x + \cos x, x \in \left[ 0, \frac{\pi}{2} \right]\]
\[f'\left( x \right) = \cos x - \sin x\]
\[\text { For f(x) to be increasing, we must have}\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \cos x - \sin x > 0\]
\[ \Rightarrow \sin x < \cos x\]
\[ \Rightarrow \frac{\sin x}{\cos x} < 1\]
\[ \Rightarrow \tan x < 1\]
\[ \Rightarrow x \in [0, \frac{\pi}{4}]\]
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