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I see that you've uploaded a screenshot of a mathematical problem related to analyzing the function \( P(x) = 2x^4 + 3x^3 - 20x^2 - 9x - 36 \) using a graphing calculator. It appears the problem asks for:

1. The correct graph of the function.
2. The domain of the function.
3. The local minimum points and whether any are absolute minimum points.
4. The local maximum points and whether any are absolute maximum points.

Here’s a breakdown of the parts of the problem:

- **Part (a): Determine the domain.**
  The domain of any polynomial function like this one is all real numbers, so the answer would be \( (-\infty, \infty) \).

- **Part (b): Determine all local minimum points and if any are absolute minimum points.**
  Based on the answer highlighted in green (correct), the local minimum points are \( (1.87, -78.69) \) and \( (-2.78, -110.55) \), with \( (-2.78, -110.55) \) being the absolute minimum point.

- **Part (c): Determine all local maximum points and if any are absolute maximum points.**
  This section is still incomplete in the image, so further input is needed to determine the correct local maximum point.

Would you like help calculating the local maximum points or need further explanation on any part of the solution?

Here are some related questions to explore:

1. How do we calculate the critical points of a polynomial function?
2. What are the steps to identify whether a point is a local or absolute minimum/maximum?
3. How do the first and second derivatives help in finding minima and maxima?
4. What is the general shape of a quartic function like \( P(x) \) and why does it have multiple extrema?
5. How can a graphing calculator be used effectively to analyze functions?

**Tip:** When dealing with higher-degree polynomials, use the derivative to find critical points and evaluate them to distinguish between local and absolute extrema.

For the function P(x) = 2x^4 + 3x^3 - 20x^2 - 9x - 36, determine the correct graph, domain, local minimum points, and local maximum points.

Explore how to find the graph, domain, local minima, and maxima for the polynomial function P(x) = 2x^4 + 3x^3 - 20x^2 - 9x - 36. This comprehensive problem involves finding extrema using derivative tests and analyzing a graph, suitable for students at the high school and college level.

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