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

Answer 1

To find the number of turning points for the function \( g(x) = 3x^3(x - 5)^2(5 - x)^4 \), we first need to identify the degree of the polynomial and the behavior of the function at its critical points.

1. **Determine the degree of the polynomial:**
- \( 3x^3 \) contributes a degree of 3.
- \( (x - 5)^2 \) contributes a degree of 2.
- \( (5 - x)^4 \) contributes a degree of 4.

So, the total degree of the polynomial is:
\[
3 + 2 + 4 = 9
\]

2. **Finding the maximum number of turning points:**
The maximum number of turning points a polynomial can have is one less than its degree. Therefore, for a polynomial of degree 9, the maximum number of turning points is:
\[
9 - 1 = 8
\]

Thus, Rogelio's sketch of the graph of \( g(x) \) should include **at most 8 turning points**.

So, the correct answer is: **eight at most**.