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!