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Question
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
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Solution
\[f\left( x \right) = x + \cos x + ax + b\]
\[f'\left( x \right) = 1 - \sin x + a\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 1 - \sin x + a > 0\]
\[ \Rightarrow \sin x < 1 + a\]
\[\text { We know that the maximum value of sin x is 1 }.\]
\[ \Rightarrow 1 + a > 1\]
\[ \Rightarrow a > 0\]
\[ \Rightarrow a \in \left( 0, \infty \right)\]
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