Question

The number of solutions of the equation $2\cos \left(\frac{x}{2}\right)=5^x+5^{-x}$ is

• A. $1$
• B. $2$
• C. $3$
• D. None of these.

Answer 1

The correct option is A $1$

Given equation $2\cos \left(\frac{x}{2}\right)=5^x+5^{-x}$
For all real values of x, $5^x+5^{-x}\ge 2$
$2\cos \left(\frac{x}{2}\right)\ge 2$
$\Rightarrow \cos \left(\frac{x}{2}\right)\ge 1$
$\Rightarrow \cos \left(\frac{x}{2}\right)=1(as\cos \left(\frac{x}{2}\right)\ngtr 1)$ $\Rightarrow \frac{x}{2}=2n\pi +0,n\in I$
$\Rightarrow x=4n\pi$
Putting $n=0,1,2,....$ we have
$x=0,4\pi ,8\pi ,......$ but $x=0$ is only the value which satisfies the equation $5^x+5^{-x}=2\cos \frac{x}{2}$
$\therefore$ Number of solution is only one i.e., $x=0$