The equation `x^(3) - 3x + 1 = 0` has

Options

A. atleast one root in `(-1,0)`
B. alteast one root in `(0,1)`
C. alteast two roots in `(-1,1)`
D. no roots in `(-1,1)`

Answer

To determine the roots of the equation \( x^3 - 3x + 1 = 0 \) and to find out if it has roots in the specified intervals, we can use the Intermediate Value Theorem. This theorem states that if a continuous function changes signs over an interval, then there is at least one root in that interval.

### Step-by-Step Solution:

1. **Define the function**:
Let \( f(x) = x^3 - 3x + 1 \).

2. **Evaluate the function at the endpoints of the intervals**:
- For the interval \([-1, 0]\):
- Calculate \( f(-1) \):
\[ f(-1) = (-1)^3 - 3(-1) + 1 = -1 + 3 + 1 = 3 \]
- Calculate \( f(0) \):
\[ f(0) = (0)^3 - 3(0) + 1 = 1 \]
- Since \( f(-1) = 3 \) and \( f(0) = 1 \), both values are positive. Thus, there is no root in the interval \([-1, 0]\).

3. **Check the interval \([0, 1]\)**:
- Calculate \( f(1) \):
\[ f(1) = (1)^3 - 3(1) + 1 = 1 - 3 + 1 = -1 \]
- We have \( f(0) = 1 \) (positive) and \( f(1) = -1 \) (negative). Since the function changes sign, there is at least one root in the interval \([0, 1]\).

4. **Check the interval \([-1, 1]\)**:
- We already know \( f(-1) = 3 \) (positive) and \( f(1) = -1 \) (negative). Since the function changes sign, there is at least one root in the interval \([-1, 1]\).

5. **Determine the number of roots**:
- To find the total number of roots, we can check the derivative \( f'(x) \):
\[ f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x - 1)(x + 1) \]
- The critical points are \( x = -1 \) and \( x = 1 \).
- Evaluating \( f(x) \) at these points, we find:
- \( f(-1) = 3 \) (local maximum)
- \( f(1) = -1 \) (local minimum)
- Since \( f(x) \) goes from positive to negative, and the behavior of the cubic function indicates that there are three roots in total, we conclude that there are at least two roots in the interval \([-1, 1]\).

### Conclusion:
The equation \( x^3 - 3x + 1 = 0 \) has at least one root in the interval \([0, 1]\) and at least two roots in the interval \([-1, 1]\).