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. (-2,0)
B. (0,2)
C. `(2, infty)`
D. `(-2, 0 )∪(2, infty)`

Answer

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

### Step 1: Define the function and its properties
Given:
\[ f(x) = 1 - x - x^3 \]
We need to find \(f'(x)\) to determine if \(f(x)\) is increasing or decreasing.

### Step 2: Calculate the derivative
The derivative of \(f(x)\) is:
\[ f'(x) = -1 - 3x^2 \]
Since \(f'(x) < 0\) for all \(x \in \mathbb{R}\), this means that \(f(x)\) is strictly decreasing.

### Step 3: Substitute \(f(x)\) into the inequality
We rewrite the inequality:
\[ 1 - f(x) - f^3(x) > f(1 - 5x) \]
First, we need to find \(f(f(x))\):
\[ f(f(x)) = f(1 - x - x^3) = 1 - (1 - x - x^3) - (1 - x - x^3)^3 \]
However, for simplicity, we will directly analyze the inequality without calculating \(f(f(x))\).

### Step 4: Analyze the inequality
We can substitute \(f(x)\) into the inequality:
\[ 1 - (1 - x - x^3) - (1 - x - x^3)^3 > f(1 - 5x) \]
This simplifies to:
\[ x + x^3 - (1 - x - x^3)^3 > f(1 - 5x) \]

### Step 5: Find \(f(1 - 5x)\)
Now we need to compute \(f(1 - 5x)\):
\[ f(1 - 5x) = 1 - (1 - 5x) - (1 - 5x)^3 \]
This simplifies to:
\[ f(1 - 5x) = 5x - (1 - 5x)^3 \]

### Step 6: Set up the inequality
Now we can set up the inequality:
\[ x + x^3 - (1 - x - x^3)^3 > 5x - (1 - 5x)^3 \]

### Step 7: Solve the inequality
We need to solve:
\[ x^3 - 4x > 0 \]
Factoring gives us:
\[ x(x^2 - 4) > 0 \]
This factors further to:
\[ x(x - 2)(x + 2) > 0 \]

### Step 8: Determine the intervals
To find the intervals where the product is positive, we analyze the sign changes:
- The critical points are \(x = -2\), \(x = 0\), and \(x = 2\).
- Testing intervals:
- For \(x < -2\): All factors are negative, product is negative.
- For \(-2 < x < 0\): One factor is negative, product is positive.
- For \(0 < x < 2\): Two factors are negative, product is negative.
- For \(x > 2\): All factors are positive, product is positive.

### Step 9: Conclusion
Thus, the solution to the inequality \(x(x - 2)(x + 2) > 0\) is:
\[ x \in (-2, 0) \cup (2, \infty) \]