Question

The value of 'a' if $f\left(x\right)=a{e}^x-a{e}^{-x}+\left(2a+1\right)x-3$ increase $\forall x$ are

• A. $\left[0,\infty \right)$
• B. $\left(-\infty ,0\right]$
• C. $\left(-\infty ,\infty \right)$
• D. $\left(1,\infty \right)$

Answer 1

The correct option is A $\left[0,\infty \right)$

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

for $f\left(x\right)$ to increase $\forall x,\frac{df\left(x\right)}{dx}>0$

$a{e}^x+a{e}^{-x}+(2a+1)>0$

$⟹a>\frac{-1}{2+e^x+e^{-x}}$

but $2+a{e}^x+a{e}^{-x}>0,\forall x⟹a\ge 0⟹a\in [0,\infty )$