Question

If \( f(x) = \frac{k \sin x + 2 \cos x}{\sin x + \cos x} \) is strictly increasing for all real values of \( x \), then: 

• A. \( k = 1 \)
• B. \( k > 1 \)
• C. \( k < 2 \)
• D. \( k > 2 \)

Hint:

Differentiate the function using the quotient rule and require the derivative to be positive for all x.

Answer 1

The Correct Option is D

Step 1: Rewrite
Divide numerator and denominator by $\cos x$: $f=\dfrac{k\tan x+2}{\tan x+1}$, a function of $t=\tan x$.

Step 2: Derivative in t
$\dfrac{df}{dt}=\dfrac{k(t+1)-(kt+2)}{(t+1)^2}=\dfrac{k-2}{(t+1)^2}$, and $\dfrac{dt}{dx}=\sec^2x\gt0$.

Step 3: Conclusion
The sign of $f'$ is the sign of $k-2$, so strict increase needs $k\gt2$.

Final Answer:
The condition is k greater than 2. \[ \boxed{\text{(D) }k \gt 2} \]