Question

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

• A. $\lambda \gt 2$
• B. $\lambda \lt 4$
• C. $\lambda \ge 4$
• D. $\lambda \gt 1$

Hint:

For any function of the form $f(x) = \frac{a \sin x + b \cos x}{c \sin x + d \cos x}$, the numerator of its derivative simplifies elegantly to a constant determinant value: $ad - bc$. For the function to be increasing, simply set $ad - bc \ge 0$. Here, $(\lambda)(3) - (6)(2) \ge 0 \implies 3\lambda - 12 \ge 0 \implies \lambda \ge 4$. This bypasses long differentiation entirely!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need the condition on λ so that f(x) = (λ sin x + 6 cos x)/(2 sin x + 3 cos x) is an increasing function.

Step 2: Key Formula or Approach:
A function is increasing if f'(x) ≥ 0. Using the quotient rule, the condition simplifies to the numerator of the derivative being non-negative.

Step 3: Detailed Explanation:
Differentiating and simplifying, the numerator reduces to 3λ(sin²x+cos²x) – 12(sin²x+cos²x) = 3λ – 12. For f'(x) ≥ 0, we need 3λ – 12 ≥ 0 → λ ≥ 4.

Step 4: Final Answer:
The function is increasing if λ ≥ 4, matching option (C).