Question

The number of real solutions of the equation $\sin \left(e^x\right)=5^x+5^{-x}$ is

• A. $1$
• B. $0$
• C. $2$
• D. $3$

Answer 1

The correct option is B

$0$

Explanation of correct answer

Given that $\sin \left(e^x\right)=5^x+5^{-x}$

We know that $AM\ge GM$

$$\begin{gathered} 5^x+5^{-x}\ge 2\sqrt{5^x·5^{-x}} \\ \Rightarrow 5^x+5^{-x}\ge 2 \end{gathered}$$

But $\left|\sin \left(e^x\right)\right|\le 1$

Therefore, there exist no solution.

Hence, the correct option is $\left(\mathrm{B}\right)$.