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

Answer

To solve the problem step by step, we start with the given information and apply the necessary mathematical principles.

### Step 1: Understand the given function
We are given that \( F(x) = \int_0^x f(t) \, dt \) and \( F(x^2) = x^2(1+x) \).

### Step 2: Differentiate \( F(x^2) \)
To find \( f(x) \), we will differentiate \( F(x^2) \) using the chain rule.

\[ \frac{d}{dx} F(x^2) = F'(x^2) \cdot \frac{d}{dx}(x^2) \]

### Step 3: Differentiate the right side
Now we differentiate the right side \( x^2(1+x) \):

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

### Step 4: Set the derivatives equal
From the differentiation we have:

\[ F'(x^2) \cdot 2x = 2x + 3x^2 \]

### Step 5: Solve for \( F'(x^2) \)
Now, we can isolate \( F'(x^2) \):

\[ F'(x^2) = \frac{2x + 3x^2}{2x} = 1 + \frac{3x}{2} \]

### Step 6: Relate \( F'(x) \) to \( f(x) \)
Since \( F'(x) = f(x) \), we can substitute \( x^2 \) back into our expression:

\[ f(x^2) = 1 + \frac{3x}{2} \]

### Step 7: Substitute \( x = 2 \) to find \( f(4) \)
Now we need to find \( f(4) \). Since \( 4 = 2^2 \), we substitute \( x = 2 \):

\[ f(4) = 1 + \frac{3 \cdot 2}{2} = 1 + 3 = 4 \]

### Final Result
Thus, the value of \( f(4) \) is:

\[ \boxed{4} \]