Question

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

• A. $0$
• B. $1$
• C. $2$
• D. None of these

Answer 1

The correct option is A $0$

$Let{y}_1=\sin \left(e^x\right)y_2=5^x+5^{-x}Now,0<e^x<\infty \therefore -1\le \sin \left(e^x\right)\le 1\therefore {\left(y_1\right)}_{max}=1\therefore {\left(y_1\right)}_{min}=-1y_2=5^x+5^{-x}=5^x+\frac{1}{5^x}Now,5^x>0Let{5}^x=t\therefore y_2=t+\frac{1}{t}Now,t+\frac{1}{t}\ge 2ift>0Hence,y_2\ge 2⟹{\left(y_2\right)}_{min}=2Therefore,y_1\ne y_2$

$\therefore$ No real solutions of the equation.

$\sin \left(e^x\right)=5^x+5^{-x}$ has zero solution.