https://file.aiquickdraw.com/mathbot/file/1726608287937w0owuv4i.jpg

Let's analyze the extrema (maximums and minimums) from the provided graphs for each function:

### 11. \( f(x) = 2x^2 + 4x - 5 \)
- This is a quadratic function (parabola), opening upwards, indicating a **minimum** point.
- **Minimum**: The vertex is the lowest point of the parabola.
- The coordinates of the minimum point appear to be around \((-1, -7)\) based on the graph.

### 12. \( f(x) = x^3 + 6x^2 + 9x + 5 \)
- This cubic function has both a **local maximum** and a **local minimum**.
- The graph shows:
  - **Maximum**: Approximately at \((-3, 8)\).
  - **Minimum**: Approximately at \((0, 5)\).

### 13. \( f(x) = -x^5 + 4x^3 - 2 \)
- This is a polynomial function of odd degree, indicating no absolute extrema but local extrema may exist.
- The graph shows:
  - **Maximum**: Approximately at \((-2, 30)\).
  - **Minimum**: Approximately at \((1, -3)\).

### 14. \( f(x) = x^4 - 4x^3 + 9x \)
- This quartic function shows both a **local maximum** and a **local minimum**.
- The graph shows:
  - **Maximum**: Approximately at \((-1, 12)\).
  - **Minimum**: Approximately at \((3, 0)\).

Would you like any specific details or further clarification?

### 5 Follow-Up Questions:
1. How can we calculate extrema using calculus (i.e., derivative-based methods)?
2. What role do the degrees of polynomials play in determining the general shape of their graphs?
3. How can we approximate extrema points without using a graphing calculator?
4. Can a function have more than one global extremum? Why or why not?
5. How do the leading coefficients of polynomials affect the direction in which the graph opens?

### Tip:
When determining extrema using a graph, always check for turning points where the slope changes from positive to negative (for maxima) or from negative to positive (for minima).

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

Learn how to identify and classify the extrema (maximums and minimums) for polynomial functions based on their graphs. This problem includes quadratic, cubic, quartic, and quintic functions, and demonstrates local and global extrema classification.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation