Question

The number of real solutions of the equation $sin\left(e^x\right)={2018}^x+{2018}^{-x}is$

• A. $0$
• B. $1$
• C. $2$
• D. $infinitely$ $many$

Answer 1

Answer: (A)

$0$

The given equation is $\sin \left(e^x\right)={2018}^x+{2018}^{-x}$

LHS $=\sin \left(e^x\right)$

Range of $\sin \left(e^x\right)$

is $(-1,1)$

RHS $={2018}^x+\frac{1}{{2018}^x}$

Using A.M$\ge G.M$ or right hand side

$\frac{{2018}^x+\frac{1}{{2018}^x}}{2}\ge \sqrt{{2018}^x\times \frac{1}{{2018}^x}}$

$\Rightarrow {2018}^x+\frac{1}{{2018}^x}\ge 2$

$\Rightarrow$ $RHS\ge 2$ for $x\in R$

by $LHS\le 1$

$\Rightarrow$ $\sin \left(e^x\right)$ cannot be equal to ${2018}^x+{2018}^{-x}$

No real solution