Question

Let \(f:\mathbf{R}\to\mathbf{R}\) be such that for all \(x\in\mathbf{R}\)
\((2^{1+x}+2^{1-x})\), \(f(x)\) and \((3^x+3^{-x})\) are in A.P.,
then the minimum value of \(f(x)\) is

• A. \(2\)
• B. \(0\)
• C. \(3\)
• D. \(4\)

Hint:

Three terms in A.P. means the middle term is the average of the other two; then apply AM-GM to each pair \(a^x+a^{-x}\ge2\).

Answer 1

The Correct Option is C

A.P. condition: \(2f(x) = (2^{1+x}+2^{1-x}) + (3^x+3^{-x})\), so \(f(x) = \frac{2(2^x+2^{-x}) + (3^x+3^{-x})}{2}\).
By AM-GM, \(2^x+2^{-x}\ge 2\) and \(3^x+3^{-x}\ge 2\), both with equality at \(x=0\).
So \(f(x)\ge \frac{2\cdot2+2}{2}=3\), with equality at \(x=0\).
Minimum value of \(f(x)\) is 3.
Answer: (3) \(3\)