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Question
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
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Solution
\[f\left( x \right) = a \left( x + \sin x \right) + a\]
\[f'\left( x \right) = a \left( 1 + \cos x \right)\]
\[\text { For }f(x)\text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow a \left( 1 + \cos x \right) > 0 . . . \left( 1 \right)\]
\[\text { We know,}\]
\[ - 1 \leq \cos x \leq 1, \forall x \in R\]
\[ \Rightarrow 0 \leq \left( 1 + \cos x \right) \leq 2, \forall x \in R\]
\[\therefore a > 0 \left[ \text { From eq }. \left( 1 \right) \right]\]
\[ \Rightarrow a \in \left( 0, \infty \right)\]
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