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Question
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
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Solution
\[f\left( x \right) = x + \cos x - a\]
\[f'\left( x \right) = 1 - \sin x\]
\[\text { We know, }\]
\[\sin x \leq 1, \forall x \in R\]
\[ \Rightarrow - \sin x \geq - 1, \forall x \in R\]
\[ \Rightarrow 1 - \sin x \geq 0, \forall x \in R\]
\[ \Rightarrow f'\left( x \right) \geq 0, \forall x \in R\]
\[\text { Hence,f }\left( x \right) \text { is increasing on R for all values of a } .\]
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