If `f (x) +2f (1 - x) = x^2 +2 AA x in R ` then f (x) is equal to

Options

A. `x^2 -2`
B. `1`
C. `1/3 (x-2)^2`
D. None of these

Answer

To solve the equation \( f(x) + 2f(1 - x) = x^2 + 2 \) for \( f(x) \), we will follow these steps:

### Step 1: Write the original equation
We start with the equation:
\[ f(x) + 2f(1 - x) = x^2 + 2 \tag{1} \]

### Step 2: Substitute \( x \) with \( 1 - x \)
Now, we substitute \( x \) with \( 1 - x \) in the original equation:
\[ f(1 - x) + 2f(x) = (1 - x)^2 + 2 \]
Expanding \( (1 - x)^2 \):
\[ f(1 - x) + 2f(x) = 1 - 2x + x^2 + 2 \]
This simplifies to:
\[ f(1 - x) + 2f(x) = x^2 - 2x + 3 \tag{2} \]

### Step 3: Set up the system of equations
Now we have two equations:
1. \( f(x) + 2f(1 - x) = x^2 + 2 \) (Equation 1)
2. \( f(1 - x) + 2f(x) = x^2 - 2x + 3 \) (Equation 2)

### Step 4: Eliminate \( f(1 - x) \)
To eliminate \( f(1 - x) \), we can multiply Equation (2) by 2:
\[ 2f(1 - x) + 4f(x) = 2(x^2 - 2x + 3) \]
This simplifies to:
\[ 2f(1 - x) + 4f(x) = 2x^2 - 4x + 6 \tag{3} \]

### Step 5: Subtract Equation (1) from Equation (3)
Now, we subtract Equation (1) from Equation (3):
\[ (2f(1 - x) + 4f(x)) - (f(x) + 2f(1 - x)) = (2x^2 - 4x + 6) - (x^2 + 2) \]
This simplifies to:
\[ 3f(x) = 2x^2 - 4x + 6 - x^2 - 2 \]
Combining like terms:
\[ 3f(x) = x^2 - 4x + 4 \]

### Step 6: Solve for \( f(x) \)
Now, divide both sides by 3:
\[ f(x) = \frac{1}{3}(x^2 - 4x + 4) \]
Factoring the quadratic:
\[ f(x) = \frac{1}{3}((x - 2)^2) \]

### Final Result
Thus, the function \( f(x) \) is:
\[ f(x) = \frac{1}{3}(x - 2)^2 \]