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