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
exactly eight
exactly eight
nine at most
nine at most
three at most
three at most
eight at most
. How many turning points should his sketch include?(1 point)
Responses
exactly eight
exactly eight
nine at most
nine at most
three at most
three at most
eight 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 first need to analyze the degrees of its polynomial components.
1. The degree of \( 3x^3 \) is 3.
2. The degree of \( (x - 5)^2 \) is 2.
3. The degree of \( (5 - x)^4 \) is 4.
Now, we sum these degrees to find the degree of the entire polynomial:
\[
3 + 2 + 4 = 9.
\]
For a polynomial of degree \( n \), the maximum number of turning points is given by \( n - 1 \). Therefore, for a polynomial of degree 9, the maximum number of turning points is:
\[
9 - 1 = 8.
\]
With this calculation in mind, the correct answer is:
**eight at most.**
1. The degree of \( 3x^3 \) is 3.
2. The degree of \( (x - 5)^2 \) is 2.
3. The degree of \( (5 - x)^4 \) is 4.
Now, we sum these degrees to find the degree of the entire polynomial:
\[
3 + 2 + 4 = 9.
\]
For a polynomial of degree \( n \), the maximum number of turning points is given by \( n - 1 \). Therefore, for a polynomial of degree 9, the maximum number of turning points is:
\[
9 - 1 = 8.
\]
With this calculation in mind, the correct answer is:
**eight at most.**