Question

Let \( f : (0, \infty) \to \mathbb{R} \) be the continuous function such that:

\[ f(x) = 2 + \frac{g(x)}{x} \quad \text{for all} \ x > 0, \quad g(x) = \int_1^x f(t) \, dt \quad \text{for all} \ x > 0. \]

Then \( f(2) \) is equal to:

• A. \( 2 + \log 2 \)
• B. \( 2 - \log 2 \)
• C. \( 2 + \log 4 \)
• D. \( 2 - \log 4 \)

Hint:

When solving for values of functions given integrals, differentiate both sides using the Fundamental Theorem of Calculus and apply the properties of derivatives.

Answer 1

The Correct Option is C

We are given that \( f(x) = 2 + \frac{g(x)}{x} \) and \( g(x) = \int_1^x f(t) \, dt \). We need to find \( f(2) \).

First, differentiate \( g(x) = \int_1^x f(t) \, dt \) using the Fundamental Theorem of Calculus:

\[ g'(x) = f(x) \]

Thus, we have the relationship \( g'(x) = f(x) \). Substitute this into the equation for \( f(x) \):

\[ f(x) = 2 + \frac{g(x)}{x} \]

Next, differentiate both sides of the equation for \( f(x) \) with respect to \( x \), using the product and quotient rules:

\[ f'(x) = -\frac{g(x)}{x^2} + \frac{f(x)}{x} \]

Now, substitute \( f(2) \) and solve to get:

\[ f(2) = 2 + \log 4 \]

Thus, the correct answer is (C) \( 2 + \log 4 \).