Question

If \[ f(x)=\frac1{x^2}\int_{3}^{x}\left(2t-3f'(t)\right)\,dt, \] then \[ f'(3)= \]

• A. \(-\dfrac12\)
• B. \(\dfrac12\)
• C. \(-\dfrac13\)
• D. \(\dfrac13\)

Hint:

When an integral equation contains \[ \int_a^x F(t)\,dt, \] first multiply away any outside factor and then differentiate using the Fundamental Theorem of Calculus: \[ \frac{d}{dx}\int_a^x F(t)\,dt = F(x). \]

Answer 1

The Correct Option is B

Concept: Use the Fundamental Theorem of Calculus and then substitute \(x=3\) to obtain an equation involving \(f'(3)\).

Step 1: Multiply both sides by \(x^2\). Given \[ f(x)=\frac1{x^2}\int_3^x(2t-3f'(t))\,dt. \] Therefore, \[ x^2f(x) = \int_3^x(2t-3f'(t))\,dt. \] \[ \cdots (1) \]

Step 2: Differentiate both sides. Differentiating (1), \[ 2xf(x)+x^2f'(x) = 2x-3f'(x). \] \[ \cdots (2) \]

Step 3: Find \(f(3)\). Substituting \(x=3\) in the original equation, \[ f(3) = \frac1{9}\int_3^3(2t-3f'(t))dt = 0. \] Thus, \[ f(3)=0. \]

Step 4: Substitute \(x=3\) into (2). Using \(f(3)=0\), \[ 2(3)(0)+9f'(3) = 2(3)-3f'(3). \] \[ 9f'(3) = 6-3f'(3). \] \[ 12f'(3)=6. \] \[ f'(3)=\frac12. \]

Step 5: Write the final answer. \[ \boxed{\frac12} \]