#### Question

Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?

#### 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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Solution Find the Set of Values of 'B' for Which F(X) = B (X + Cos X) + 4 is Decreasing on R ? Concept: Increasing and Decreasing Functions.