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Question
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
Solution
\[f\left( x \right) = \sin x - bx + c\]
\[f'\left( x \right) = \cos x - b\]
\[\text { Given }:f\left( x \right) \text { is decreasing on R }.\]
\[f'\left( x \right) < 0, \forall x \in R\]
\[ \Rightarrow \cos x - b < 0, \forall x \in R\]
\[\Rightarrow\cos x - b < 0, \forall x \in R \]
\[ \Rightarrow \cos x < b, \forall x \in R\]
\[ \Rightarrow b \geqslant 1 \left[ \because - 1 \leqslant \cos x \leqslant 1 \right]\]
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