Let's go through the questions based on the image you've provided.

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 $(2.5, -20)$.

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 $(-1, 1)$.

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.