Question

Rogelio is asked to sketch a graph of g(x)=3x3(x−5)2(5−x)4. How many turning points should his sketch include?(1 point) Responses eight at most eight at most nine at most nine at most exactly eight exactly eight three at most

Answers

GPT-4o mini
To determine the maximum number of turning points for the function \( g(x) = 3x^3(x - 5)^2(5 - x)^4 \), we can use the fact that for a polynomial of degree \( n \), the maximum number of turning points is \( n - 1 \).

First, we need to find the degree of the polynomial \( g(x) \):

- The term \( 3x^3 \) contributes a degree of 3.
- The term \( (x - 5)^2 \) contributes a degree of 2.
- The term \( (5 - x)^4 \) contributes a degree of 4.

Adding these contributions together gives us the total degree:
\[
3 + 2 + 4 = 9
\]

Since the degree of the polynomial \( g(x) \) is 9, the maximum number of turning points is:
\[
9 - 1 = 8
\]

Thus, Rogelio's sketch should include **exactly eight** turning points.
Are you sure
GPT-4o mini
Let's verify the calculation step by step to ensure accuracy regarding the polynomial \( g(x) = 3x^3(x - 5)^2(5 - x)^4 \).

1. **Identify the degrees of each factor**:
- The factor \( 3x^3 \) has a degree of **3**.
- The factor \( (x - 5)^2 \) has a degree of **2**.
- The factor \( (5 - x)^4 \) has a degree of **4**.

2. **Total degree**:
The total degree of \( g(x) \) is the sum of the degrees of its factors:
\[
3 + 2 + 4 = 9
\]

3. **Maximum number of turning points**:
The maximum number of turning points for a polynomial of degree \( n \) is given by \( n - 1 \). Since the degree of \( g(x) \) is 9, we calculate the maximum number of turning points as follows:
\[
9 - 1 = 8
\]

4. **Conclusion**:
So, the function \( g(x) = 3x^3(x - 5)^2(5 - x)^4 \) can indeed have a maximum of **8** turning points.

Given this confirmation, the correct answer is indeed that Rogelio's sketch should include **exactly eight** turning points.

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