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Question
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
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Solution
\[f\left( x \right) = x^2 - x \sin x\]
\[f'\left( x \right) = 2x - x \cos x - \sin x\]
\[\text { Here,} \]
\[0 < x < \frac{\pi}{2}\]
\[ \Rightarrow 0 < \sin x < 1 \text { and }0 < \cos x < 1\]
\[ \Rightarrow 2x - x \cos x - \sin x > 0\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in \left( 0, \frac{\pi}{2} \right)\]
\[\text { So },f\left( x \right)\text { is increasing on}\left( 0, \frac{\pi}{2} \right).\]
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