Let `f (x) =(1)/(x ^(2)) int _(4)^(x) (4t ^(2) -2 f '(t) dt, ` find `9f'(4)`

Options

A. 16
B. 4
C. 8
D. 32

Answer

To solve the problem, we need to find \( 9f'(4) \) given the function:

\[ f(x) = \frac{1}{x^2} \int_{4}^{x} (4t^2 - 2f'(t)) dt \]

### Step 1: Rewrite the function \( f(x) \)

We start with the function:

\[ f(x) = \frac{1}{x^2} \int_{4}^{x} (4t^2 - 2f'(t)) dt \]

### Step 2: Split the integral

We can split the integral into two parts:

\[ f(x) = \frac{1}{x^2} \left( \int_{4}^{x} 4t^2 dt - 2 \int_{4}^{x} f'(t) dt \right) \]

### Step 3: Evaluate the first integral

The first integral can be evaluated as follows:

\[ \int_{4}^{x} 4t^2 dt = \left[ \frac{4t^3}{3} \right]_{4}^{x} = \frac{4x^3}{3} - \frac{4 \cdot 4^3}{3} \]

Calculating \( 4 \cdot 4^3 \):

\[ 4 \cdot 4^3 = 4 \cdot 64 = 256 \]

Thus,

\[ \int_{4}^{x} 4t^2 dt = \frac{4x^3}{3} - \frac{256}{3} \]

### Step 4: Substitute back into \( f(x) \)

Now substituting back into \( f(x) \):

\[ f(x) = \frac{1}{x^2} \left( \frac{4x^3}{3} - \frac{256}{3} - 2 \int_{4}^{x} f'(t) dt \right) \]

### Step 5: Differentiate \( f(x) \)

Next, we differentiate \( f(x) \) using the product rule:

\[ f'(x) = \frac{d}{dx} \left( \frac{1}{x^2} \right) \left( \frac{4x^3}{3} - \frac{256}{3} - 2 \int_{4}^{x} f'(t) dt \right) + \frac{1}{x^2} \frac{d}{dx} \left( \frac{4x^3}{3} - \frac{256}{3} - 2 \int_{4}^{x} f'(t) dt \right) \]

Calculating the derivative of \( \frac{1}{x^2} \):

\[ \frac{d}{dx} \left( \frac{1}{x^2} \right) = -\frac{2}{x^3} \]

### Step 6: Evaluate \( f'(4) \)

To find \( f'(4) \), we substitute \( x = 4 \) into the differentiated expression. However, we also know that:

\[ f(4) = 0 \]

This implies that the term involving \( f'(t) \) cancels out when evaluated at \( x = 4 \).

### Step 7: Solve for \( f'(4) \)

From the earlier steps, we can set up the equation for \( f'(4) \):

\[ f'(4) + \frac{1}{8} f'(4) = 4 \]

This simplifies to:

\[ \frac{9}{8} f'(4) = 4 \]

### Step 8: Isolate \( f'(4) \)

Now, solving for \( f'(4) \):

\[ f'(4) = 4 \cdot \frac{8}{9} = \frac{32}{9} \]

### Step 9: Calculate \( 9f'(4) \)

Finally, we calculate:

\[ 9f'(4) = 9 \cdot \frac{32}{9} = 32 \]

Thus, the final answer is:

\[ \boxed{32} \]