The point x=−1
 is Answer 1 Question 1
 
The point x=0
 is Answer 2 Question 1
 
The point x=2
 is Answer 3 Question 1
 
The point x=3
 is Answer 4 Question 1
 
The point x=4
 is

Let's analyze each of the points \( x = -1 \), \( x = 0 \), \( x = 2 \), \( x = 3 \), and \( x = 4 \) for the function \( f(x) = 3x^4 - 16x^3 + 6x^2 + 72x - 7 \).

### 1. \( x = -1 \)
We previously determined that:

- \( f'(-1) = 0 \)
- \( f''(-1) = 144 > 0 \)

This means \( x = -1 \) is a **local minimum**.

### 2. \( x = 0 \)
To find out whether \( x = 0 \) is a critical point, we need to check the first derivative:

\[
f'(0) = 12(0)^3 - 48(0)^2 + 12(0) + 72 = 72 \neq 0
\]

Since \( f'(0) \) is not zero, \( x = 0 \) is **not a critical point**.

### 3. \( x = 2 \)
We previously determined that:

- \( f'(2) = 0 \)
- \( f''(2) = -36 < 0 \)

This means \( x = 2 \) is a **local maximum**.

### 4. \( x = 3 \)
We previously determined that:

- \( f'(3) = 0 \)
- \( f''(3) = 48 > 0 \)

This means \( x = 3 \) is a **local minimum**.

### 5. \( x = 4 \)
Let's check if \( x = 4 \) is a critical point by finding the value of the first derivative at \( x = 4 \):

\[
f'(4) = 12(4)^3 - 48(4)^2 + 12(4) + 72
\]

Calculating:

\[
f'(4) = 12(64) - 48(16) + 48 + 72 = 768 - 768 + 120 = 120 \neq 0
\]

Since \( f'(4) \) is not zero, \( x = 4 \) is **not a critical point**.

### Summary:
- The point \( x = -1 \) is a **local minimum**.
- The point \( x = 0 \) is **not a critical point**.
- The point \( x = 2 \) is a **local maximum**.
- The point \( x = 3 \) is a **local minimum**.
- The point \( x = 4 \) is **not a critical point**.

Would you like more details or have any questions about this analysis? Here are some related questions to consider:

1. How would you find the critical points of a trigonometric function?
2. What is the importance of the second derivative in graphing functions?
3. How do you interpret the results of the first derivative test?
4. Can a point where the derivative is undefined be a critical point?
5. What happens if both first and second derivatives are zero at a point?
6. How can inflection points be determined?
7. How does the degree of the polynomial affect the number of critical points?
8. What is the significance of concavity in determining maxima and minima?

**Tip:** Checking the sign of the first derivative across intervals is useful for confirming whether a point is indeed a maximum or minimum.

Explore the analysis of critical points and extrema for the polynomial function f(x) = 3x^4 - 16x^3 + 6x^2 + 72x - 7 at points x = -1, x = 0, x = 2, x = 3, and x = 4. This detailed analysis covers the application of the first derivative test and second derivative test, highlighting local minima and maxima. Suitable for advanced high school students.

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