The domain of `f (x) = log_(2) (2x^(3) - x^(2) - 4x + 2)`, is

Options

A. ` (-sqrt2, 1//2) uu ( sqrt2, oo)`
B. `(-1, 1//2) uu ( sqrt2, oo)`
C. `(-sqrt2,-1 ) uu (1,oo)`
D. ` (-oo, -sqrt2)uu( sqrt2, oo)`

Explanation

To find the domain of the function \( f(x) = \log_2(2x^3 - x^2 - 4x + 2) \), we need to determine where the argument of the logarithm is positive, since logarithms are only defined for positive values.

### Step-by-Step Solution:

1. **Set the argument of the logarithm greater than zero**:
\[ 2x^3 - x^2 - 4x + 2 > 0 \]

2. **Factor the polynomial**:
We will first try to factor the polynomial \( 2x^3 - x^2 - 4x + 2 \). We can group terms:
\[ 2x^3 - 4x - x^2 + 2 = 2x(x^2 - 2) - 1(x^2 - 2) \]
Factoring out \( (x^2 - 2) \):
\[ = (2x - 1)(x^2 - 2) \]

3. **Further factor \( x^2 - 2 \)**:
The expression \( x^2 - 2 \) can be factored as:
\[ x^2 - 2 = (x - \sqrt{2})(x + \sqrt{2}) \]
Therefore, we can write:
\[ 2x^3 - x^2 - 4x + 2 = (2x - 1)(x - \sqrt{2})(x + \sqrt{2}) \]

4. **Determine the critical points**:
The critical points from the factors are:
\[ 2x - 1 = 0 \implies x = \frac{1}{2} \]
\[ x - \sqrt{2} = 0 \implies x = \sqrt{2} \]
\[ x + \sqrt{2} = 0 \implies x = -\sqrt{2} \]

5. **Test intervals around the critical points**:
We will test the sign of \( (2x - 1)(x - \sqrt{2})(x + \sqrt{2}) \) in the intervals:
- \( (-\infty, -\sqrt{2}) \)
- \( (-\sqrt{2}, \frac{1}{2}) \)
- \( (\frac{1}{2}, \sqrt{2}) \)
- \( (\sqrt{2}, \infty) \)

- **Interval \( (-\infty, -\sqrt{2}) \)**: Choose \( x = -2 \):
\[ (2(-2) - 1)(-2 - \sqrt{2})(-2 + \sqrt{2}) \Rightarrow (-5)(-2 - \sqrt{2})(-2 + \sqrt{2}) > 0 \]
(Positive)

- **Interval \( (-\sqrt{2}, \frac{1}{2}) \)**: Choose \( x = 0 \):
\[ (2(0) - 1)(0 - \sqrt{2})(0 + \sqrt{2}) \Rightarrow (-1)(-\sqrt{2})(\sqrt{2}) < 0 \]
(Negative)

- **Interval \( (\frac{1}{2}, \sqrt{2}) \)**: Choose \( x = 1 \):
\[ (2(1) - 1)(1 - \sqrt{2})(1 + \sqrt{2}) \Rightarrow (1)(1 - \sqrt{2})(1 + \sqrt{2}) < 0 \]
(Negative)

- **Interval \( (\sqrt{2}, \infty) \)**: Choose \( x = 2 \):
\[ (2(2) - 1)(2 - \sqrt{2})(2 + \sqrt{2}) \Rightarrow (3)(2 - \sqrt{2})(2 + \sqrt{2}) > 0 \]
(Positive)

6. **Combine the intervals where the expression is positive**:
The expression \( 2x^3 - x^2 - 4x + 2 > 0 \) is positive in the intervals:
\[ (-\infty, -\sqrt{2}) \cup (\sqrt{2}, \infty) \]

### Conclusion:
Thus, the domain of the function \( f(x) = \log_2(2x^3 - x^2 - 4x + 2) \) is:
\[ (-\infty, -\sqrt{2}) \cup (\sqrt{2}, \infty) \]