Question

If \[ 3f(x) - 2f\left(\frac{1}{x}\right) = x, \] then \( f'(2) \) is:

• A. \( 1 \)
• B. \( \frac{1}{2} \)
• C. \( 2 \)
• D. \( \frac{7}{2} \)

Hint:

For functional equations, assume a linear form and solve using coefficient comparison.

Answer 1

The Correct Option is B

Step 1: Finding \( f(x) \) Assume \( f(x) = ax + b \). Substituting: \[ 3(ax + b) - 2(a/x + b) = x. \] Expanding and equating coefficients: \[ 3ax + 3b - 2a/x - 2b = x. \] Solving, we get: \[ a = \frac{1}{2}, \quad b = 0. \] Thus, \[ f(x) = \frac{x}{2}. \] Step 2: Finding \( f'(x) \) \[ f'(x) = \frac{1}{2}, \quad f'(2) = \frac{1}{2}. \]