A function `f(x)` satisfies the functional equation `x^2f(x)+f(1-x)=2x-x^4` for all real `x. f(x)` must be

Options

A. `x^(2)`
B. `1-x^(2)`
C. `1+x^(2)`
D. `x^(2)+x+1`

Explanation

To solve the functional equation \( x^2 f(x) + f(1-x) = 2x - x^4 \) for all real \( x \), we can follow these steps:

### Step 1: Substitute \( x \) with \( 1-x \)
We start by substituting \( x \) with \( 1-x \) in the original equation:

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

### Step 2: Simplify the equation
Now, we simplify the right-hand side:

\[ 2(1-x) = 2 - 2x \]
\[ (1-x)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4 \]

So, the equation becomes:

\[ (1-x)^2 f(1-x) + f(x) = 2 - 2x - (1 - 4x + 6x^2 - 4x^3 + x^4) \]

This simplifies to:

\[ (1-x)^2 f(1-x) + f(x) = 2 - 2x - 1 + 4x - 6x^2 + 4x^3 - x^4 \]
\[ = 1 + 2x - 6x^2 + 4x^3 - x^4 \]

### Step 3: Write down the two equations
Now we have two equations:

1. \( x^2 f(x) + f(1-x) = 2x - x^4 \) (Equation 1)
2. \( (1-x)^2 f(1-x) + f(x) = 1 + 2x - 6x^2 + 4x^3 - x^4 \) (Equation 2)

### Step 4: Express \( f(1-x) \) from Equation 1
From Equation 1, we can express \( f(1-x) \):

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

### Step 5: Substitute \( f(1-x) \) into Equation 2
Now, substitute \( f(1-x) \) into Equation 2:

\[ (1-x)^2 (2x - x^4 - x^2 f(x)) + f(x) = 1 + 2x - 6x^2 + 4x^3 - x^4 \]

### Step 6: Expand and simplify
Expanding the left-hand side:

\[ (1-x)^2 (2x - x^4) - (1-x)^2 x^2 f(x) + f(x) = 1 + 2x - 6x^2 + 4x^3 - x^4 \]

### Step 7: Collect like terms
Now, collect like terms and isolate \( f(x) \):

\[ (1-x)^2 (2x - x^4) + f(x)(1 - (1-x)^2 x^2) = 1 + 2x - 6x^2 + 4x^3 - x^4 \]

### Step 8: Solve for \( f(x) \)
Rearranging gives us:

\[ f(x) = \frac{(1 + 2x - 6x^2 + 4x^3 - x^4) - (1-x)^2 (2x - x^4)}{1 - (1-x)^2 x^2} \]

### Step 9: Simplify the expression
After simplifying the expression, we find:

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

### Conclusion
Thus, the function \( f(x) \) that satisfies the given functional equation is:

\[ \boxed{1 - x^2} \]