Question

Function $f(x) = \frac{\lambda\sin x + 6\cos x}{2\sin x + 3\cos x}$ is monotonic increasing if

• A. $\lambda > 1$
• B. $\lambda < 1$
• C. $\lambda < 4$
• D. $\lambda > 4$

Answer 1

Answer: (D)

$\lambda > 4$

Answer 2

The function is monotonic increasing if, $f^{'}(x) > 0$

⇒$\frac{(2\sin x + 3\cos x)(\lambda\cos x - 6\sin x)}{(2\sin x + 3\cos x)^{2}}$– $\frac{(\lambda\sin x + 6\cos x)(2\cos x - 3\sin x)}{(2\sin x + 3\cos x)^{2}} > 0$

⇒ $3\lambda(\sin^{2}x + \cos^{2}x) - 12(\sin^{2}x + \cos^{2}x) > 0$ ⇒ $3\lambda - 12 > 0$

⇒ $\lambda > 4$.