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The problem asks to find the minimum value of the function \( f(x) = x^3 - 3x - 7 \) using Newton's Method, with an initial approximation \( x_1 = 0.5 \) and a tolerance \( \epsilon = 0.001 \).

### Newton's Method for Finding a Minimum
Newton's method for finding minima involves using the first and second derivatives of the function:
- \( f'(x) \) (first derivative) gives the slope.
- \( f''(x) \) (second derivative) is used to update the approximation.

The Newton's method iteration for optimization is:
\[
x_{n+1} = x_n - \frac{f'(x_n)}{f''(x_n)}
\]
where:
- \( f'(x) = 3x^2 - 3 \)
- \( f''(x) = 6x \)

We'll apply this iterative process starting with \( x_1 = 0.5 \) until the change in \( x \) is less than the given tolerance \( \epsilon = 0.001 \).

Let me calculate this step-by-step.Using Newton's Method, the minimum value of \( f(x) = x^3 - 3x - 7 \) occurs approximately at \( x \approx 1.0003 \) after 3 iterations, with the given initial approximation \( x_1 = 0.5 \) and tolerance \( \epsilon = 0.001 \).

Would you like more details on the process or have any questions?

Here are five related questions to further explore:
1. How would Newton's method change if the initial approximation was different?
2. Can you apply Newton's method to maximize the function instead?
3. What is the role of the second derivative in Newton's method for finding minima?
4. How does the convergence of Newton's method depend on the function's curvature?
5. How would the results differ if the tolerance \( \epsilon \) were smaller or larger?

**Tip**: Always ensure that the second derivative is non-zero when applying Newton's method for optimization.

Find minimum value of f(x) = x^3 - 3x - 7, using Newton’s Method taking initial approximation x1 = 0.5 and ε = 0.001.

Learn how to find the minimum value of the function f(x) = x^3 - 3x - 7 using Newton’s Method. This problem involves key calculus concepts, such as derivatives and optimization, with an initial approximation of x1 = 0.5 and tolerance ε = 0.001. A detailed step-by-step approach to solving this problem is provided, suitable for university-level students.

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