For `x in R`, the functions f(x) satisfies `2f(x)+f(1-x)=x^(2)`. The value of f(4) is equal to

Options

A. `13/3`
B. `43/3`
C. `23/3`
D. None of these

Answer

To solve the problem, we need to find the value of \( f(4) \) given the functional equation:

\[ 2f(x) + f(1-x) = x^2 \]

### Step 1: Substitute \( x = 4 \)

We start by substituting \( x = 4 \) into the functional equation:

\[ 2f(4) + f(1-4) = 4^2 \]

This simplifies to:

\[ 2f(4) + f(-3) = 16 \tag{1} \]

### Step 2: Substitute \( x = -3 \)

Next, we substitute \( x = -3 \) into the functional equation:

\[ 2f(-3) + f(1 - (-3)) = (-3)^2 \]

This simplifies to:

\[ 2f(-3) + f(4) = 9 \tag{2} \]

### Step 3: Set up the system of equations

Now we have two equations:

1. \( 2f(4) + f(-3) = 16 \) (Equation 1)
2. \( 2f(-3) + f(4) = 9 \) (Equation 2)

### Step 4: Solve the system of equations

Let's denote \( f(4) = x \) and \( f(-3) = y \). We can rewrite the equations as:

1. \( 2x + y = 16 \)
2. \( 2y + x = 9 \)

Now we can solve this system of equations.

From Equation 1, we can express \( y \) in terms of \( x \):

\[ y = 16 - 2x \tag{3} \]

Now substitute Equation (3) into Equation (2):

\[ 2(16 - 2x) + x = 9 \]

Expanding this gives:

\[ 32 - 4x + x = 9 \]

Combining like terms:

\[ 32 - 3x = 9 \]

### Step 5: Isolate \( x \)

Now, isolate \( x \):

\[ -3x = 9 - 32 \]
\[ -3x = -23 \]
\[ x = \frac{23}{3} \]

### Step 6: Find \( f(4) \)

Since we defined \( x = f(4) \), we have:

\[ f(4) = \frac{23}{3} \]

### Final Answer

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

\[ \boxed{\frac{23}{3}} \]