Statement I `If f (x) = int_(0)^(1) (xf(t)+1) dt,` then `int_(0)^(3) f (x) dx =12`
Statement II `f(x)=3x+1`

Options

A. Statement I is true, Statement II is also true, Statement II is the correct explanation of Statement 1.
B. Statement I is true, Statement II is also true , Statement II is not the correct explanation of Statement II.
C. Statement I is true, Statement II is false
D. Statement I is false , Statement II is true

Answer

To solve the problem step by step, we will analyze the statements and derive the necessary equations.

### Step 1: Understand the given function
We are given the function:
\[ f(x) = \int_{0}^{1} (x f(t) + 1) \, dt \]

### Step 2: Split the integral
We can split the integral into two parts:
\[ f(x) = \int_{0}^{1} x f(t) \, dt + \int_{0}^{1} 1 \, dt \]

### Step 3: Evaluate the second integral
The second integral is straightforward:
\[ \int_{0}^{1} 1 \, dt = [t]_{0}^{1} = 1 - 0 = 1 \]

### Step 4: Factor out the constant
Since \( x \) is constant with respect to \( t \), we can factor it out of the first integral:
\[ f(x) = x \int_{0}^{1} f(t) \, dt + 1 \]

Let \( k = \int_{0}^{1} f(t) \, dt \). Thus, we can rewrite:
\[ f(x) = kx + 1 \]

### Step 5: Integrate \( f(x) \) from 0 to 3
Now we need to evaluate:
\[ \int_{0}^{3} f(x) \, dx = \int_{0}^{3} (kx + 1) \, dx \]

### Step 6: Compute the integral
Calculating the integral:
\[ \int_{0}^{3} (kx + 1) \, dx = \int_{0}^{3} kx \, dx + \int_{0}^{3} 1 \, dx \]

The first integral:
\[ \int_{0}^{3} kx \, dx = k \left[ \frac{x^2}{2} \right]_{0}^{3} = k \left( \frac{3^2}{2} - 0 \right) = \frac{9k}{2} \]

The second integral:
\[ \int_{0}^{3} 1 \, dx = [x]_{0}^{3} = 3 - 0 = 3 \]

Combining these results:
\[ \int_{0}^{3} f(x) \, dx = \frac{9k}{2} + 3 \]

### Step 7: Set the equation equal to 12
According to the problem statement:
\[ \frac{9k}{2} + 3 = 12 \]

### Step 8: Solve for \( k \)
Subtract 3 from both sides:
\[ \frac{9k}{2} = 12 - 3 = 9 \]
Multiply both sides by 2:
\[ 9k = 18 \]
Divide by 9:
\[ k = 2 \]

### Step 9: Substitute back to find \( f(x) \)
Now substitute \( k \) back into the equation for \( f(x) \):
\[ f(x) = 2x + 1 \]

### Step 10: Verify Statement II
Statement II claims \( f(x) = 3x + 1 \). Since we found \( f(x) = 2x + 1 \), Statement II is false.

### Conclusion
Thus, Statement I is true, and Statement II is false.