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

For $f(x) = \frac{k \sin x + 2 \cos x}{\sin x + \cos x}$ to be strictly increasing, its derivative must be positive for all $x$ where defined.

Differentiating using the quotient rule, let $u = k \sin x + 2 \cos x$ with $u' = k \cos x - 2 \sin x$, and $v = \sin x + \cos x$ with $v' = \cos x - \sin x$. Then:

$f'(x) = \frac{(k \cos x - 2 \sin x)(\sin x + \cos x) - (k \sin x + 2 \cos x)(\cos x - \sin x)}{(\sin x + \cos x)^2}$

Expanding and simplifying, the numerator reduces to $k(\sin^2 x + \cos^2 x) - 2(\sin^2 x + \cos^2 x) = k - 2$.

Hence, $f'(x) = \frac{k - 2}{(\sin x + \cos x)^2}$.

The denominator is always positive when defined, so $f'(x) > 0$ only when $k - 2 > 0$, i.e., $k > 2$.

The function is strictly increasing for all real $x$ (where defined) if and only if $k > 2$.