The largest interval for which `x^(12) - x^(9) + x^(4) - x+1 gt 0` is

Options

A. `-4 lt x le 0`
B. `0 lt x lt 1`
C. `-100 lt x lt 100`
D. `-oo lt x lt oo`

Answer

To solve the inequality \( x^{12} - x^{9} + x^{4} - x + 1 > 0 \), we will analyze the expression step by step.

### Step 1: Analyze the expression
We start with the expression:
\[ f(x) = x^{12} - x^{9} + x^{4} - x + 1 \]
We need to find the intervals where \( f(x) > 0 \).

### Step 2: Test values in different intervals
We will test specific values of \( x \) to see if the inequality holds.

#### Testing \( x = -5 \)
Substituting \( x = -5 \):
\[ f(-5) = (-5)^{12} - (-5)^{9} + (-5)^{4} - (-5) + 1 \]
Calculating each term:
- \( (-5)^{12} = 244140625 \)
- \( -(-5)^{9} = 1953125 \)
- \( (-5)^{4} = 625 \)
- \( -(-5) = 5 \)
- \( 1 = 1 \)

Putting it all together:
\[ f(-5) = 244140625 + 1953125 + 625 + 5 + 1 = 246093376 \]
Since \( 246093376 > 0 \), the inequality holds for \( x = -5 \).

#### Testing \( x = 0 \)
Substituting \( x = 0 \):
\[ f(0) = 0^{12} - 0^{9} + 0^{4} - 0 + 1 = 1 \]
Since \( 1 > 0 \), the inequality holds for \( x = 0 \).

#### Testing \( x = 1 \)
Substituting \( x = 1 \):
\[ f(1) = 1^{12} - 1^{9} + 1^{4} - 1 + 1 = 1 - 1 + 1 - 1 + 1 = 1 \]
Since \( 1 > 0 \), the inequality holds for \( x = 1 \).

#### Testing \( x = 100 \)
Substituting \( x = 100 \):
\[ f(100) = 100^{12} - 100^{9} + 100^{4} - 100 + 1 \]
Calculating:
- \( 100^{12} \) is a very large number.
- \( -100^{9} \) is also large but smaller than \( 100^{12} \).
- \( 100^{4} = 100000000 \)
- \( -100 + 1 = -99 \)

Since \( 100^{12} \) dominates the expression, \( f(100) > 0 \).

### Step 3: Conclusion
Since we tested various values and found that \( f(x) > 0 \) for all of them, we can conclude that the inequality holds for all \( x \) in the interval \( (-\infty, \infty) \).

Thus, the largest interval for which \( x^{12} - x^{9} + x^{4} - x + 1 > 0 \) is:
\[ \boxed{(-\infty, \infty)} \]