https://file.aiquickdraw.com/mathbot/file/1726785727036t9muiirk.jpg

Let's analyze each function's extrema by examining the graphs provided in the image:

### Problem 11: \( f(x) = 2x^2 + 4x - 5 \)
This is a quadratic function (a parabola) opening upwards, meaning it has a **minimum** point. Based on the graph, the vertex (minimum) seems to be around \( (-1, -7) \).

- **Extrema Type**: Minimum
- **Coordinates**: Approximately \( (-1, -7) \)

### Problem 12: \( f(x) = x^3 + 6x^2 + 9x + 5 \)
This cubic function appears to have one local maximum and one local minimum. The local maximum seems to occur around \( (-3, 14) \), and the local minimum around \( (0, 5) \).

- **Extrema Types**: 
  - Local Maximum at \( (-3, 14) \)
  - Local Minimum at \( (0, 5) \)

### Problem 13: \( f(x) = -x^5 + 4x^3 - 2 \)
This is a degree 5 polynomial, which appears to have both a local maximum and a local minimum. The local maximum is around \( (-1, 1) \), and the local minimum is around \( (1, -1) \).

- **Extrema Types**: 
  - Local Maximum at \( (-1, 1) \)
  - Local Minimum at \( (1, -1) \)

### Problem 14: \( f(x) = x^4 - 4x^3 + 9x \)
This quartic function has a local maximum and a local minimum. The local maximum seems to be around \( (-1, 12) \), and the local minimum around \( (2, -8) \).

- **Extrema Types**: 
  - Local Maximum at \( (-1, 12) \)
  - Local Minimum at \( (2, -8) \)

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Do you want further details on how to classify the extrema mathematically or any specific explanations on how these values are estimated?

Here are 5 related questions:
1. How do you find the exact coordinates of extrema using calculus?
2. How do we confirm if a point is a maximum or minimum?
3. What role does the derivative play in finding extrema?
4. How do the degree of a polynomial and the shape of its graph relate?
5. How can we find inflection points in polynomials?

**Tip**: When working with higher-degree polynomials, their graphs can have multiple turning points. Use the first and second derivatives to identify these points more precisely.

Give the coordinates and classify the extrema for the graph of each function.

Learn how to classify the extrema (minimums and maximums) for polynomial functions using graphing and calculus techniques. Explore cubic and quartic functions, including key points of extrema and their coordinates.

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