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The given function is \( y = x^3 - 2x^2 - 5x + 2 \). We found the critical points by solving the first derivative and determining the concavity with the second derivative. Here are the results:

- The critical points are \( x \approx -0.786 \) and \( x \approx 2.120 \).
- The second derivative test shows that at \( x \approx -0.786 \), the second derivative is negative, indicating a local maximum. At \( x \approx 2.120 \), the second derivative is positive, indicating a local minimum.
- Evaluating the function at these points gives approximate values of:
  - \( y(-0.786) \approx 4.209 \) (local maximum)
  - \( y(2.120) \approx -8.061 \) (local minimum)

Therefore, the local maximum occurs at the point \( (-0.786, 4.209) \), which corresponds to option (D).

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. How do you find critical points of a function?
2. What does the second derivative test indicate?
3. What is the significance of concavity in determining local maxima or minima?
4. How would you graph a cubic function like this one?
5. How can you confirm whether a point is a local maximum or minimum graphically?

**Tip**: Always check the second derivative to confirm whether a critical point is a maximum or minimum!

The graph of y = x^3 - 2x^2 - 5x + 2 has a local maximum at:

Determine the local maximum of the cubic function y = x^3 - 2x^2 - 5x + 2 by finding the critical points and applying the second derivative test. Suitable for students in Grades 11-12 studying calculus and polynomial functions.

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