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

To solve the problem, we need to evaluate the given integral condition: \(\int_0^x t f(t) \, dt = x^2 f(x)\) for \(x > 0\) and find \( f(6) \). First, differentiate both sides of the equation with respect to \(x\). The left side is an integral of the form \(\int_0^x u f(u) \, du\), which can be differentiated using the Leibniz rule for differentiation under the integral sign:

\(\frac{d}{dx}\left(\int_0^x tf(t) \, dt\right) = x f(x) + \int_0^x t \frac{d}{dx}f(t) \, dt\).

The right side of the equation differentiated with respect to \(x\) gives:

\(\frac{d}{dx}(x^2 f(x)) = 2x f(x) + x^2 f'(x)\).

Equating the derivatives of both sides:

\(x f(x) + \int_0^x t f'(t) \, dt = 2x f(x) + x^2 f'(x).\)

By comparing both sides, \(\int_0^x t f'(t) \, dt\) needs to satisfy \(x f(x) = 2x f(x) + x^2 f'(x)\). 

This simplifies to:

\(0 = x f(x) + x^2 f'(x)\).

Rearranging terms gives us:

\(x f(x)(1 + x f'(x)/f(x)) = 0\).

Since \(x > 0\), it implies:

\(f(x) + x f'(x) = 0\), such that \(f(x)\) satisfies the differential equation \(f'(x)/f(x) = -1/x\).

This differential equation can be solved by separating variables:

\(\frac{d}{dx}\ln|f(x)| = -\frac{1}{x}\).

Integrate both sides:

\(\ln|f(x)| = -\ln|x| + C\) where \(C\) is the constant of integration.

\(|f(x)| = \frac{e^C}{x}\) gives our function \(f(x) = \frac{k}{x}\) where \(k = e^C\) is a constant.

Since \( f(2) = 3 \), substitute into the function:

\(f(2) = \frac{k}{2} = 3\) which implies \(k = 6\).

Therefore, the function is \(f(x) = \frac{6}{x}\).

Finally, we compute \( f(6) \): \(f(6) = \frac{6}{6} = 1\).