Let `f(x)=x^(12)-x^(9)+x^(4)-x+1`. Which of the following is true ?

Options

A. f is one-one
B. f has a real root
C. f' never vabishes
D. f take only positive values

Explanation

To solve the problem, we need to analyze the function \( f(x) = x^{12} - x^{9} + x^{4} - x + 1 \) and determine which of the provided statements is true.

### Step-by-Step Solution:

1. **Identify the Function**:
We start with the function:
\[ f(x) = x^{12} - x^{9} + x^{4} - x + 1 \]

2. **Evaluate the Function at Specific Points**:
- **Evaluate \( f(0) \)**:
\[ f(0) = 0^{12} - 0^{9} + 0^{4} - 0 + 1 = 1 \]
- **Evaluate \( f(1) \)**:
\[ f(1) = 1^{12} - 1^{9} + 1^{4} - 1 + 1 = 1 - 1 + 1 - 1 + 1 = 1 \]
- **Evaluate \( f(-1) \)**:
\[ f(-1) = (-1)^{12} - (-1)^{9} + (-1)^{4} - (-1) + 1 = 1 + 1 + 1 + 1 + 1 = 5 \]

3. **Analyze the Values**:
From the evaluations:
- \( f(0) = 1 \) (positive)
- \( f(1) = 1 \) (positive)
- \( f(-1) = 5 \) (positive)

All evaluated points yield positive values.

4. **Determine if \( f(x) \) is Always Positive**:
Since we have found that \( f(0) \), \( f(1) \), and \( f(-1) \) are all positive, we can hypothesize that \( f(x) \) may be positive for all \( x \).

5. **Check for Roots**:
Since \( f(0) = 1 \) and \( f(1) = 1 \), and both are positive, we can infer that there are no real roots in the interval [0, 1].

6. **Consider the Derivative**:
To check if \( f(x) \) is one-to-one, we can look at the derivative \( f'(x) \). If \( f'(x) \) is always positive or always negative, then \( f(x) \) is either strictly increasing or strictly decreasing, respectively.

7. **Conclusion**:
Since we have established that \( f(x) \) is positive for the evaluated points and likely for all \( x \), we conclude that:
\[ \text{The correct statement is: } f(x) \text{ takes only positive values.} \]

### Final Answer:
The true statement is: **Option 4: \( f \) takes only positive values.**