https://file.aiquickdraw.com/mathbot/file/1727069431203pha3hwnv.jpg

The problem asks to find the value of \( x \) where the local minimum of the function \( f(x) = x^3 - 3x + 2 \) occurs.

### Steps to solve:

1. **Find the derivative of \( f(x) \):**
   The derivative \( f'(x) \) helps identify critical points where the slope of the function is zero (possible local minima or maxima).
   \[
   f'(x) = \frac{d}{dx}(x^3 - 3x + 2) = 3x^2 - 3
   \]

2. **Set the derivative equal to zero:**
   To find critical points:
   \[
   3x^2 - 3 = 0
   \]
   Simplify:
   \[
   x^2 = 1 \quad \Rightarrow \quad x = \pm 1
   \]

3. **Determine the nature of the critical points:**
   Use the second derivative \( f''(x) \) to check whether each critical point is a local minimum or maximum.
   
   The second derivative is:
   \[
   f''(x) = \frac{d}{dx}(3x^2 - 3) = 6x
   \]
   - For \( x = 1 \):
     \[
     f''(1) = 6(1) = 6 \quad (\text{positive, so local minimum at } x = 1)
     \]
   - For \( x = -1 \):
     \[
     f''(-1) = 6(-1) = -6 \quad (\text{negative, so local maximum at } x = -1)
     \]

Thus, the local minimum occurs at \( x = 1 \).

The correct response is **Response 2: \( x = 1 \)**.

---

Would you like further details on any steps or concepts involved in this solution?

Here are five related questions to deepen your understanding:
1. How do second derivatives help in identifying local minima and maxima?
2. What are the necessary conditions for a critical point to be a saddle point?
3. Can a cubic function have more than one local minimum?
4. How does concavity relate to the second derivative of a function?
5. How would the solution change if the function had higher-order terms?

**Tip**: Always check critical points using the second derivative test to confirm whether they are minima, maxima, or saddle points.

Given the function f(x) = x^3 - 3x + 2, find the value of x where the local minimum occurs.

Learn how to find the local minimum of the cubic function f(x) = x^3 - 3x + 2 using derivatives. This problem involves finding critical points and applying the second derivative test to determine if a point is a local minimum. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation