Question

If the function $f\left(x\right)=\frac{k\sin x+2\cos x}{\sin x+\cos x}$ is strictly increasing for all values of $x$ in its domain, then

• A. $k<1$
• B. $k>1$
• C. $k<2$
• D. $k>2$

Answer 1

The correct option is D $k>2$

Since $f\left(x\right)=\frac{k\sin x+2\cos x}{\sin x+\cos x}$
$\therefore f^{\prime }\left(x\right)>0$ ($\because f\left(x\right)$ is strictly increasing for all $x$ in its domain).

$f^{\prime }\left(x\right)=\frac{(\sin x+\cos x)(k\cos x-2\sin x)-(k\sin x+2\cos x)(\cos x-\sin x)}{(\sin x+\cos x{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{k({\sin }^{2}x+{\cos }^{2}x)-2({\sin }^{2}x+{\cos }^{2}x)}{(\sin x+\cos x{)}^2}$
$\Rightarrow \frac{k-2}{(\sin x+\cos x{)}^2}>0$ for all $x$ in its domain.
$\Rightarrow (k-2)>0\Rightarrow k>2$