`x^(8)-x^(5)-(1)/(x)+(1)/(x^(4)) gt 0`, is satisfied for

Options

A. only positive values of x
B. only negative values of x
C. all real numbers except zero
D. only for `x gt 1`

Answer

To solve the inequality \( x^8 - x^5 - \frac{1}{x} + \frac{1}{x^4} > 0 \), we will analyze the function step by step.

### Step 1: Rewrite the inequality
We start with the function:
\[ f(x) = x^8 - x^5 - \frac{1}{x} + \frac{1}{x^4} \]
We need to determine when \( f(x) > 0 \).

### Step 2: Test specific values
Let's evaluate \( f(x) \) at some specific values to understand the behavior of the function.

#### Test \( x = -1 \):
\[ f(-1) = (-1)^8 - (-1)^5 - \frac{1}{-1} + \frac{1}{(-1)^4} \]
Calculating this gives:
\[ f(-1) = 1 + 1 + 1 + 1 = 4 > 0 \]
So, \( f(-1) > 0 \).

#### Test \( x = 1 \):
\[ f(1) = 1^8 - 1^5 - \frac{1}{1} + \frac{1}{1^4} \]
Calculating this gives:
\[ f(1) = 1 - 1 - 1 + 1 = 0 \]
So, \( f(1) = 0 \).

#### Test \( x = 2 \):
\[ f(2) = 2^8 - 2^5 - \frac{1}{2} + \frac{1}{2^4} \]
Calculating this gives:
\[ f(2) = 256 - 32 - 0.5 + 0.0625 \]
\[ = 256 - 32 - 0.5 + 0.0625 = 223.5625 > 0 \]
So, \( f(2) > 0 \).

### Step 3: Analyze the function near \( x = 0 \)
As \( x \) approaches 0, \( f(x) \) becomes undefined due to the terms \( -\frac{1}{x} \) and \( \frac{1}{x^4} \). Therefore, we need to consider the behavior of \( f(x) \) as \( x \) approaches 0 from the left and right.

- As \( x \to 0^+ \), \( -\frac{1}{x} \to -\infty \) and \( \frac{1}{x^4} \to +\infty \), making \( f(x) \to +\infty \).
- As \( x \to 0^- \), \( -\frac{1}{x} \to +\infty \) and \( \frac{1}{x^4} \to +\infty \), making \( f(x) \to +\infty \).

### Step 4: Determine intervals of positivity
From our tests:
- \( f(-1) > 0 \)
- \( f(1) = 0 \)
- \( f(2) > 0 \)

We can conclude:
- \( f(x) > 0 \) for \( x < 0 \) (since \( f(-1) > 0 \)).
- \( f(x) > 0 \) for \( x > 1 \) (since \( f(2) > 0 \)).
- \( f(x) = 0 \) at \( x = 1 \).

### Conclusion
The function \( f(x) > 0 \) is satisfied for:
\[ x < 0 \quad \text{and} \quad x > 1 \]
Thus, the solution to the inequality is:
\[ (-\infty, 0) \cup (1, \infty) \]