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

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.**

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