Question

Find the possible values of $a$ such that $f\left(x\right)=e^{2x}-(a+1)e^x+2x$ is monoatomically increasing for $x\in R$.

Answer 1

$f\left(x\right)=e^{2x}-(a+1)e^x+2x$
$\Rightarrow f^{\prime }\left(x\right)=2e^{2x}-(a+1)e^x+2$
Now, $2e^{2x}-(a+1)e^x+2\ge 0$ for all $x\in R$
$\Rightarrow 2(e^x+\frac{1}{e^x})-(a+1)\ge 0$ for all $x\in R$
$\Rightarrow \frac{a+1}{2}\le 4(\because e^x+\frac{1}{e^x}hasminimumvalue2)$