#### Question

Function *f*(*x*) = cos *x* − 2 λ *x* is monotonic decreasing when

(a) λ > 1/2

(b) λ < 1/2

(c) λ < 2

(d) λ > 2

#### Solution

\[f\left( x \right) = \cos x - 2 \lambda x\]

\[f'\left( x \right) = - \sin x - 2 \lambda \]

\[\text { For f(x) to be decreasing, we must have }\]

\[f'\left( x \right) < 0\]

\[ \Rightarrow - \sin x - 2 \lambda < 0\]

\[ \Rightarrow sin x + 2 \lambda > 0 \]

\[ \Rightarrow 2 \lambda > - \sin x\]

\[\text { We know that the maximum value of -sin x is 1 }.\]

\[ \Rightarrow 2 \lambda > 1\]

\[ \Rightarrow \lambda > \frac{1}{2}\]

Is there an error in this question or solution?

Solution Function F(X) = Cos X − 2 λ X is Monotonic Decreasing When (A) λ > 1/2 (B) λ < 1/2 (C) λ < 2 (D) λ > 2 Concept: Increasing and Decreasing Functions.