Verify that the root of the equation x³ - 3x + 1 = 0 lies between 0 and 0.8

To verify that the root of the equation \( x^3 - 3x + 1 = 0 \) lies between 0 and 0.8, we need to check the values of the function \( f(x) = x^3 - 3x + 1 \) at the endpoints of the interval. If the function changes sign between these points, then by the Intermediate Value Theorem, there is at least one root in this interval.

1. Calculate \( f(0) \):
   \[
   f(0) = 0^3 - 3 \cdot 0 + 1 = 1
   \]

2. Calculate \( f(0.8) \):
   \[
   f(0.8) = (0.8)^3 - 3 \cdot 0.8 + 1 = 0.512 - 2.4 + 1 = -0.888
   \]

Now we observe:
- \( f(0) = 1 \) (positive)
- \( f(0.8) = -0.888 \) (negative)

Since \( f(0) \) is positive and \( f(0.8) \) is negative, the function \( f(x) = x^3 - 3x + 1 \) changes sign between \( x = 0 \) and \( x = 0.8 \). By the Intermediate Value Theorem, there is at least one root of the equation \( x^3 - 3x + 1 = 0 \) in the interval \( (0, 0.8) \).

Therefore, we have verified that a root of the equation lies between 0 and 0.8.

Would you like any additional details or have any questions?

Here are some related questions for further exploration:
1. What is the exact value of the root in the interval \( (0, 0.8) \)?
2. How many roots does the equation \( x^3 - 3x + 1 = 0 \) have in total?
3. Can you find another interval where another root of \( x^3 - 3x + 1 = 0 \) lies?
4. How would you use Newton's method to approximate the root in the interval \( (0, 0.8) \)?
5. What does the graph of \( y = x^3 - 3x + 1 \) look like?
6. How would you determine the intervals where the function \( f(x) = x^3 - 3x + 1 \) is increasing or decreasing?
7. What are the critical points of the function \( f(x) = x^3 - 3x + 1 \)?
8. How do you determine the concavity and inflection points of the function \( f(x) = x^3 - 3x + 1 \)?

**Tip:** When verifying roots of a polynomial equation within a certain interval, always check the signs of the function values at the endpoints and apply the Intermediate Value Theorem for a thorough validation.

Learn how to verify that the root of the equation x^3 - 3x + 1 = 0 lies between 0 and 0.8 using the Intermediate Value Theorem. This problem involves evaluating the function f(x) = x^3 - 3x + 1 at specified endpoints to ensure a root exists within the given interval. Suitable for advanced high school students.

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