Question

{If \(f(x)\) is a quadratic function such that \(f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{1-x}\right)\), then \(\sqrt{f\left(\frac{2}{3}\right) + f\left(\frac{3}{2}\right)} =\)}

• A. \(\frac{25}{12}\)
• B. \(\frac{10}{3}\)
• C. \(\frac{13}{6}\)
• D. \(\frac{41}{20}\)

Hint:

Utilize the symmetry properties of quadratic functions to simplify calculations, especially when dealing with function values at symmetric points around 1.

Answer 1

The Correct Option is C

The given function satisfies the condition \(f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{1-x}\right)\). Given this property, to solve for \(\sqrt{f\left(\frac{2}{3}\right) + f\left(\frac{3}{2}\right)}\), we first determine the values of \(f\left(\frac{2}{3}\right)\) and \(f\left(\frac{3}{2}\right)\) by applying quadratic properties and symmetry. After computing these values and their sum, we find the square root of their sum to be \(\frac{13}{6}\).