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Question
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Options
λ > 1/2
λ < 1/2
λ < 2
λ > 2
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Solution
\[f\left( x \right) = \cos x - 2 \lambda x\]
\[f'\left( x \right) = - \sin x - 2 \lambda \]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow - \sin x - 2 \lambda < 0\]
\[ \Rightarrow sin x + 2 \lambda > 0 \]
\[ \Rightarrow 2 \lambda > - \sin x\]
\[\text { We know that the maximum value of -sin x is 1 }.\]
\[ \Rightarrow 2 \lambda > 1\]
\[ \Rightarrow \lambda > \frac{1}{2}\]
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