Question

The number of the real roots of the equation 2cos(x(x+1)) = 2^x + 2^-x is 

• A. 2
• B. 1
• C. infinite
• D. 0

Answer 1

The Correct Option is B

For any real value of x , the expression 2cos(x(x+1)) = 2^x + 2^-x  would always be positive.
Now, to find the maximum value of 2cos(x(x+1)) = 2^x + 2^-x.
we can apply the AM-GM inequality.
2^x + 2^-x / 2 ≥ √2^x + 2^-x
⇒ 2^x + 2^-x  ≥ 2√2⁰
⇒ 2^x + 2^-x ≥ 2
∴ 2cos(x(x+1)) ≥ 2
It is known that -1 ≤ cosθ ≤ 1
⇒ 2cos(x(x+1)) = 2
Therefore, the expression is valid only if 2^x + 2^-x = 2 , which is true for only one value of x i.e. 0.
 ∴ expression has only one real solution.