Question

If $f\left(x\right)=2e^x-a{e}^{-x}+(2a+1)x-3$ is increasing for all values of x then?

• A. $a\in (-\infty ,\infty )$
• B. $a\in (-\infty ,0]$
• C. $a\in [0,\infty )$
• D. None of these

Answer 1

Answer: (B)

$a\in (-\infty ,0]$

For increasing function, $f^{\prime }\left(x\right)>0$

$f\left(x\right)=2e^x-a{e}^{-x}+(2a+1)x-3$

$⟹f^{\prime }\left(x\right)=\frac{2e^x}{increase}+\frac{a{e}^{-x}}{decrease}+(2a+a)>0$

So, $a$ must be $-ve$ i.e. $a\in (-\infty ,0]$