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Question
Show that f(x) = x − sin x is increasing for all x ∈ R ?
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Solution
\[f\left( x \right) = x - \sin x\]
\[f'\left( x \right) = 1 - \cos x\]
\[\text { For f(x) to be increasing, we must have}\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 1 - \cos x > 0\]
\[ \Rightarrow f'(x) \geqslant 0 \text { for all } x \in R \left[ \because Cos x \leqslant 1 \right]\]
\[\text { So, f(x) is increasing for all } x \in R . \]
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