Question

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

exactly eight
exactly eight

eight at most
eight at most

nine at most
nine at most

three at most

Answer 1

To determine the number of turning points of the function \( g(x) = 3x^3(x-5)^2(5-x)^4 \), we will first analyze the factors in the polynomial.

1. **Identify the Degree of the Polynomial:**
The polynomial \( g(x) \) is a product of the following terms:
- \( 3x^3 \) (degree 3)
- \( (x-5)^2 \) (degree 2)
- \( (5-x)^4 \) (degree 4)

To find the total degree of \( g(x) \):
\[
\text{Degree}(g(x)) = 3 + 2 + 4 = 9
\]

2. **Determine the Number of Turning Points:**
The maximum number of turning points for a polynomial is given by \( \text{degree} - 1 \). Therefore, for \( g(x) \):
\[
\text{Maximum Turning Points} = 9 - 1 = 8
\]

3. **Evaluate the Multiplicities:**
The turning points occur at the critical points where the derivative \( g'(x) \) changes sign. The multiplicities of the roots can affect whether turning points occur or not.

- The \( x^3 \) term introduces a root at \( x = 0 \) with an odd multiplicity, leading to one turning point.
- The \( (x-5)^2 \) term introduces a root at \( x = 5 \) with an even multiplicity, which does not contribute a turning point, as it does not change the sign.
- The \( (5-x)^4 \) term at \( x = 5 \) (again due to even multiplicity) does not contribute a turning point.

4. **Conclusion:**
Since there is one turning point at \( x = 0 \), and no additional contributions from \( x = 5 \) (due to even multiplicities), the function can have **at most 8 turning points** due to its total degree.

Thus, the correct response is:
- **eight at most**.