Let `f(x)=1-x-x^3`.Find all real values of x satisfying the inequality, `1-f(x)-f^3(x)>f(1-5x)`

Options

A. `(-oo, -2)`
B. `(-2, 0) uu (2, oo)`
C. `(1, 2)`
D. `(0, 2)`

Answer

To solve the inequality \( 1 - f(x) - f^2(x) > f(1 - 5x) \) where \( f(x) = 1 - x - x^3 \), we will follow these steps:

### Step 1: Substitute \( f(x) \) into the inequality
We start by substituting \( f(x) \) into the inequality:

\[ 1 - (1 - x - x^3) - (1 - x - x^3)^2 > f(1 - 5x) \]

### Step 2: Simplify \( 1 - f(x) \)
Calculating \( 1 - f(x) \):

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

### Step 3: Calculate \( f^2(x) \)
Next, we need to calculate \( f^2(x) \):

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

Expanding this:

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

### Step 4: Substitute \( f(1 - 5x) \)
Now we need to calculate \( f(1 - 5x) \):

\[ f(1 - 5x) = 1 - (1 - 5x) - (1 - 5x)^3 \]

Calculating \( (1 - 5x)^3 \):

\[ (1 - 5x)^3 = 1 - 15x + 75x^2 - 125x^3 \]

So,

\[ f(1 - 5x) = 1 - (1 - 5x) - (1 - 15x + 75x^2 - 125x^3) = 5x + 15x - 75x^2 + 125x^3 \]

### Step 5: Combine everything into the inequality
We can now rewrite the inequality:

\[ x + x^3 - (1 - 2x + x^2 - 2x^3 + 2x^4 + x^6) > 5x + 15x - 75x^2 + 125x^3 \]

### Step 6: Rearranging the inequality
Rearranging gives:

\[ x + x^3 - 1 + 2x - x^2 + 2x^3 - 2x^4 - x^6 > 20x - 75x^2 + 125x^3 \]

### Step 7: Collect like terms
Combining like terms leads to:

\[ -x^6 + 2x^4 + (x^3 - 125x^3) + (x^2 + 75x^2) + (x + 2x - 20x - 1) > 0 \]

### Step 8: Solve the resulting polynomial inequality
This will lead to a polynomial inequality that we can solve for \( x \).

### Final Step: Determine the intervals
After solving the polynomial inequality, we will find the intervals of \( x \) that satisfy the inequality.