Question

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

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

Answer 1

The correct option is A $0$

$\sin \left(e^x\right)=5^x+5^{-x}$
we know that, $a+b\ge 2\sqrt{ab}\forall a,b\ge 0$
$\Rightarrow 5^x+5^{-x}\ge 2\sqrt{5^x\cdot 5^{-x}}\ge 2$
So, R.H.S.$\ge 2$
But L.H.S. $=\sin \left(e^x\right)\le 1$
L.H.S. $\le 1$ and R.H.S.$\ge 2$
So, there is no real solution.