Increasing Functions

Since $f(x)=e^{2x}-(k+1)e^x+2x$ , therefore $f'(x) =2e^{2x}-(k+1)e^x+2$

$=2\left (e^{2x}-\dfrac{k+1}{2}e^x+1\right )$

$=2\left (e^x-\dfrac{k+1}{4}\right )^2-\dfrac{k^2+2k-15}{8}$ . For $f(x)$ to be an increasing function, the minimum value of $f'(x)$ must be non-negative.

So $k^2+2k-15\leq 0\implies -5\leq k\leq 3$ .

Hence the maximum value of $k$ is $\boxed 3$ .