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### For Questions 11 to 14:
#### 11. \( f(x) = 2x^2 + 4x - 5 \)
This is a quadratic function that represents a parabola opening upwards (positive coefficient of \(x^2\)).
- **Vertex (Extrema):** The vertex can be found by using the formula \( x = \frac{-b}{2a} \) for quadratic equations. Here, \(a = 2\) and \(b = 4\), so:
  \[
  x = \frac{-4}{2(2)} = -1
  \]
  Substituting \(x = -1\) into the equation:
  \[
  f(-1) = 2(-1)^2 + 4(-1) - 5 = 2 - 4 - 5 = -7
  \]
  So, the vertex is at \( (-1, -7) \). Since the parabola opens upwards, this is a **minimum** point.

#### 12. \( f(x) = x^3 + 6x^2 + 9x + 5 \)
This is a cubic function. From the graph, it looks like there are two extrema: one maximum and one minimum.
- **Maximum:** Approximate from the graph, it looks like around \( (-2, 9) \).
- **Minimum:** The minimum appears around \( (0, 5) \).

#### 13. \( f(x) = -x^5 + 4x^3 - 2 \)
This is a degree 5 polynomial. The graph shows a local maximum and minimum.
- **Maximum:** Approximate from the graph, it is around \( (0, -2) \).
- **Minimum:** There’s a minimum around \( (2, -30) \) based on the graph.

#### 14. \( f(x) = x^4 - 4x^3 + 9x \)
This is a degree 4 polynomial. It seems to have a local maximum and minimum.
- **Maximum:** Approximate from the graph, it is around \( (-0.5, 5) \).
- **Minimum:** The minimum looks around \( (2.5, -20) \).

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### For Questions 15 to 20 (Intervals of Increase/Decrease):
#### 15. The graph indicates that the function is decreasing from \( (-\infty, 0) \) and increasing from \( (0, \infty) \).

#### 16. The graph shows the function decreases in \( (-\infty, 1) \) and increases from \( (1, \infty) \).

#### 17. The function is increasing on \( (-\infty, -2) \cup (2, \infty) \) and decreasing in the interval \( (-2, 2) \).

#### 18. \( f(x) = -x^5 + 3x^3 + 1 \)
- **Increasing:** In the intervals \( (-2, 0) \cup (2, \infty) \)
- **Decreasing:** In the intervals \( (-\infty, -2) \cup (0, 2) \)

#### 19. \( f(x) = -|2x - 5| + 6 \)
- **Increasing:** From \( (2.5, \infty) \)
- **Decreasing:** From \( (-\infty, 2.5) \)

#### 20. \( f(x) = x^3 + 2x^2 - 3x - 4 \)
- **Increasing:** The function increases on \( (-\infty, -1) \cup (1, \infty) \)
- **Decreasing:** It decreases on \( (-1, 1) \).

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Would you like me to provide more details or clarification on any of these problems?

Here are 5 relative questions for practice:
1. How do you determine the vertex of a quadratic function?
2. What does the degree of a polynomial tell you about the number of extrema it can have?
3. How can you find intervals of increase and decrease without using a graphing calculator?
4. What is the significance of the first derivative in finding local extrema?
5. How does the leading coefficient of a polynomial function affect its end behavior?

**Tip:** The second derivative of a function helps determine concavity, which can be useful when analyzing local maxima or minima.

Provide the coordinates and classify the extrema for the graph of each function and give the intervals on which each function increases or decreases.

This guide provides detailed solutions for identifying extrema and analyzing intervals of increase and decrease for polynomial functions. The explanations include methods like using derivatives and vertex formulas, suitable for students in advanced high school or college-level calculus.

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