Question

Let \( f(x) \) be a polynomial such that \( f(x) + f(1/x) = f(x)f(1/x) \), \( x > 0 \). If \( \int f(x)\,dx = g(x) + c \) and \( g(1) = \frac{4}{3} \), \( f(3) = 10 \), then \( g(3) \) is:

• A. 10
• B. 9
• C. 8
• D. 12

Hint:

For \(f(x) + f(1/x) = f(x)f(1/x)\), rewrite as \([f(x)-1][f(1/x)-1] = 1\). Then \(f(x)-1 = \pm x^n\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The functional equation \(f(x) + f(1/x) = f(x) \cdot f(1/x)\) for polynomials is a standard form. Its only solutions are \(f(x) = \pm x^{n} + 1\).
Step 2: Detailed Explanation:
1. Find \(f(x)\):
Given \(f(3) = 10\).
Test \(f(x) = x^{n} + 1\): \(3^{n} + 1 = 10 \implies 3^{n} = 9 \implies n = 2\).
So, \(f(x) = x^{2} + 1\).
2. Find \(g(x)\):
\[ g(x) = \int (x^{2} + 1) dx = \frac{x^{3}}{3} + x \]
Check the constant using \(g(1) = 4/3\):
\(g(1) = 1/3 + 1 = 4/3\). This matches, so the integration constant \(c\) in \(g(x)\) is zero.
3. Calculate \(g(3)\):
\[ g(3) = \frac{3^{3}}{3} + 3 = \frac{27}{3} + 3 = 9 + 3 = 12 \]
Step 3: Final Answer:
The value of \(g(3)\) is 12.