CBSE (Science) Class 12CBSE
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

Solution for The Function F ( X ) = λ Sin X + 2 Cos X Sin X + Cos X is Increasing, If (A) λ < 1 (B) λ > 1 (C) λ < 2 (D) λ > 2 - CBSE (Science) Class 12 - Mathematics

Login
Create free account


      Forgot password?

Question

The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if

(a) λ < 1
(b) λ > 1
(c) λ < 2
(d) λ > 2

Solution

(d) λ > 2

\[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\]

\[f'\left( x \right) = \frac{\left( \sin x + \cos x \right)\left( \lambda \cos x - 2 \sin x \right) + \left( \lambda \sin x + 2 \cos x \right)\left( \cos x - \sin x \right)}{\left( \sin x + \cos x \right)^2}\]

\[ = \frac{\lambda\cos x \sin x + \lambda \cos^2 x - 2 \sin^2 x - 2 \sin x \cos x - \lambda\sin x \cos x - 2 \cos^2 x + \lambda \sin^2 x + 2 \cos x \sin x}{\left( \sin x + \cos x \right)^2}\]

\[ = \frac{- 2 \left( \sin^2 x + \cos^2 x \right) + \lambda \left( \sin^2 x + \cos^2 x \right)}{\left( \sin x + \cos x \right)^2}\]

\[ = \frac{- 2 + \lambda}{\left( \sin x + \cos x \right)^2}\]

\[\text { Forf(x) to be increasing, we must have }\]

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

\[ \Rightarrow \frac{- 2 + \lambda}{\left( \sin x + \cos x \right)^2} > 0 \]

\[ \Rightarrow \lambda - 2 > 0 \left[ \because \left( \sin x + \cos x \right)^2 > 0 \right]\]

\[ \Rightarrow \lambda > 2\]

  Is there an error in this question or solution?
Solution The Function F ( X ) = λ Sin X + 2 Cos X Sin X + Cos X is Increasing, If (A) λ < 1 (B) λ > 1 (C) λ < 2 (D) λ > 2 Concept: Increasing and Decreasing Functions.
S
View in app×