If `F(x)= (1)/(x^(2)) int_(4)^(x) [4t^(2)- 2F' (t)]dt`, then F'(4) equals

Options

A. 32
B. `(32)/(3)`
C. `(32)/(9)`
D. None of these

Answer

To solve the problem, we need to find \( F'(4) \) given the expression for \( F(x) \):

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

### Step 1: Differentiate \( F(x) \) with respect to \( x \)

Using the product rule and the Fundamental Theorem of Calculus, we differentiate \( F(x) \):

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

### Step 2: Calculate \( \frac{d}{dx} \left( \frac{1}{x^2} \right) \)

The derivative of \( \frac{1}{x^2} \) is:

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

### Step 3: Substitute into the derivative expression

Now substituting this back into the expression for \( F'(x) \):

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

### Step 4: Evaluate \( F'(4) \)

To find \( F'(4) \), we substitute \( x = 4 \):

\[ F'(4) = -\frac{2}{4^3} \int_{4}^{4} \left( 4t^2 - 2F'(t) \right) dt + \frac{1}{4^2} \left( 4 \cdot 4^2 - 2F'(4) \right) \]

Since the integral from 4 to 4 is zero:

\[ F'(4) = 0 + \frac{1}{16} \left( 64 - 2F'(4) \right) \]

### Step 5: Simplify the equation

This simplifies to:

\[ F'(4) = \frac{1}{16} \cdot 64 - \frac{1}{8} F'(4) \]

\[ F'(4) = 4 - \frac{1}{8} F'(4) \]

### Step 6: Solve for \( F'(4) \)

Now, we can isolate \( F'(4) \):

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

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

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

### Final Answer
Thus, the value of \( F'(4) \) is:

\[ \boxed{\frac{32}{9}} \]