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We are tasked with finding the stationary points for the cubic function \( y = 2x^3 - 24x + 13 \), which involves finding the critical points and determining whether they are maxima or minima.

### Steps to solve:
1. **Find the first derivative** of the function to locate the stationary points.
   - Given \( y = 2x^3 - 24x + 13 \), the derivative is:
   \[
   \frac{dy}{dx} = 6x^2 - 24
   \]
   
2. **Set the derivative equal to zero** to find the x-values of the stationary points:
   \[
   6x^2 - 24 = 0
   \]
   Simplifying:
   \[
   x^2 = 4 \implies x = \pm 2
   \]

3. **Find the y-values** of the stationary points by substituting \( x = 2 \) and \( x = -2 \) into the original equation:
   - For \( x = 2 \):
     \[
     y = 2(2)^3 - 24(2) + 13 = 16 - 48 + 13 = -19
     \]
     So, one stationary point is \( (2, -19) \).
   - For \( x = -2 \):
     \[
     y = 2(-2)^3 - 24(-2) + 13 = -16 + 48 + 13 = 45
     \]
     So, the other stationary point is \( (-2, 45) \).

4. **Determine the nature of each stationary point** by analyzing the second derivative:
   - The second derivative is:
     \[
     \frac{d^2y}{dx^2} = 12x
     \]
   - For \( x = 2 \):
     \[
     \frac{d^2y}{dx^2} = 12(2) = 24 > 0
     \]
     This indicates a **local minimum** at \( (2, -19) \).
   - For \( x = -2 \):
     \[
     \frac{d^2y}{dx^2} = 12(-2) = -24 < 0
     \]
     This indicates a **local maximum** at \( (-2, 45) \).

### Final answer:
- **Maximum:** \( (-2, 45) \)
- **Minimum:** \( (2, -19) \)

Would you like further details or have any questions?

### Related Questions:
1. What does the second derivative tell us about concavity?
2. How do we find inflection points for cubic functions?
3. Why does setting the first derivative equal to zero give stationary points?
4. What is the significance of the sign of the second derivative at a critical point?
5. How would the behavior of the function change if the cubic term had a negative coefficient?

### Tip:
Always check the sign of the second derivative to confirm the nature (maximum or minimum) of a stationary point.

Find the coordinates and the nature of each of the stationary points on the curve with equation y = 2x^3 - 24x + 13.

Learn how to find and classify the stationary points of the cubic function y = 2x^3 - 24x + 13. The step-by-step solution involves finding the first and second derivatives, identifying the maximum and minimum points, and applying the second derivative test. Suitable for Grades 11-12 calculus students.

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