Let `f:(0,oo)to R and F(x)=int_(0)^(x) f(t)dt. " If " F(x^(2))=x^(2)(1+x)`, then f(4) equals

Options

A. `(5)/(4)`
B. 7
C. 4
D. 2

Explanation

To solve the problem, we start with the given function \( F(x) = \int_0^x f(t) dt \) and the condition \( F(x^2) = x^2(1+x) \). We need to find \( f(4) \).

### Step-by-Step Solution:

1. **Differentiate Both Sides**:
We start by differentiating the equation \( F(x^2) = x^2(1+x) \) with respect to \( x \).
\[ \frac{d}{dx} F(x^2) = \frac{d}{dx} [x^2(1+x)] \]

2. **Apply the Chain Rule**:
Using the chain rule on the left side, we have:
\[ \frac{d}{dx} F(x^2) = f(x^2) \cdot \frac{d}{dx}(x^2) = f(x^2) \cdot 2x \]

3. **Differentiate the Right Side**:
Now, differentiate the right side:
\[ \frac{d}{dx} [x^2(1+x)] = 2x(1+x) + x^2 = 2x + 2x^2 \]

4. **Set the Derivatives Equal**:
Now we equate both sides:
\[ f(x^2) \cdot 2x = 2x + 2x^2 \]

5. **Solve for \( f(x^2) \)**:
Dividing both sides by \( 2x \) (assuming \( x \neq 0 \)):
\[ f(x^2) = 1 + x \]

6. **Substitute \( x = 2 \)**:
We need to find \( f(4) \), which corresponds to \( f(x^2) \) when \( x = 2 \):
\[ f(4) = f(2^2) = 1 + 2 = 3 \]

### Final Answer:
Thus, the value of \( f(4) \) is \( \boxed{3} \).