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.