Question

Let for some function \( y = f(x) \), \(\int_0^x t f(t) \, dt = x^2 f(x), x>0\) and \( f(2) = 3 \). Then \( f(6) \) is equal to:

• A. 3
• B. 1
• C. 6
• D. 2

Hint:

Recognize that \( \frac{d}{dx} \int_0^x f(t) \, dt = f(x) \).

Answer 1

The Correct Option is B

Step 1: Differentiate the given equation.
Upon differentiating both sides with respect to \( x \), we get the equation \( f(x) + x f'(x) = 2x f(x) + x^2 f'(x) \).
Step 2: Solve the resulting differential equation.
After simplifying, we arrive at \( f(x) = \frac{c}{x^2} \).
By using the initial condition \( f(2) = 3 \), we determine \( c = 12 \).
Conclusion: Therefore, \( f(6) = \frac{12}{6^2} = 1 \).