Question

Find the value of $a$ if $f\left(x\right)=2e^x-a{e}^{-x}+(2a+1)x-3$ is increasing for all values of $x$.

Answer 1

For f(x) to be increasing for all x, f'(x) must be $\ge 0$ for all x

Thus $2e^x+a{e}^{-x}+2a+1\ge 0$

It can be clearly observed that for every a greater than 0 this inequality holds.

Let's check if it holds for any a less than 0.

Let x tend to minus infinity, in that case if we take a to be any negative value this relation fails , thus a can't be less than 0

So $a\in [0,\infty ]$