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Question
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
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Solution
\[f\left( x \right) = b\left( x + \cos x \right) + 4\]
\[f'\left( x \right) = b\left( 1 - \sin x \right)\]
\[\text { Given }:f(x) \text { is decreasing on R }.\]
\[ \Rightarrow f'\left( x \right) < 0\]
\[ \Rightarrow b\left( 1 - \sin x \right) < 0 . . . \left( 1 \right)\]
\[\text { We know },\]
\[\sin x \leq 1\]
\[ \Rightarrow 1 - \sin x \geq 0\]
\[ \Rightarrow b < 0 \left[ \text { Since } \left( 1 - \sin x \right) \geq 0, b\left( 1 - \sin x \right) < 0 \Rightarrow b < 0 \right]\]
\[ \Rightarrow b \in \left( - \infty , 0 \right)\]
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