If the function $f\left(x\right)=\frac{K \sin x +2 \cos x}{\sin x +\cos x}$ is increasing for all values of x then

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{ksin} \mathrm{x}+2 \cos \mathrm{x}}{\sin \mathrm{x}+\cos \mathrm{x}} \\ \Rightarrow f\left(x\right)=\frac{k \tan x+2}{\tan x+1} \\ {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)=\frac{k-2}{{\left(\tan \mathrm{x}+ 1\right)}^2}se{c}^{2}x \end{gathered}$$

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{increasing} \\ \mathrm{k}-2>0\Rightarrow \mathrm{k}>2 \end{gathered}$$