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 three at most three at most nine at most nine at most exactly eight exactly eight eight at most eight at most

Answer 1

To determine the number of turning points in the polynomial function \( g(x) = 3x^3(x - 5)^2(5 - x)^4 \), we first need to analyze its degree and critical points.

Step 1: Find the degree of the polynomial

We can find the degree of \( g(x) \) by summing the powers of \( x \) in each factor:

• From \( 3x^3 \), we get a degree of 3.
• From \( (x - 5)^2 \), we add a degree of 2.
• From \( (5 - x)^4 \) (which is the same as \( -(x - 5)^4 \)), we add a degree of 4.

So, the total degree is: \[ 3 + 2 + 4 = 9 \] Thus, \( g(x) \) is a polynomial of degree 9.

Step 2: Identify the maximum number of turning points

According to calculus, the maximum number of turning points of a polynomial function is one less than its degree. Therefore, since \( g(x) \) is of degree 9, the maximum number of turning points is: \[ 9 - 1 = 8 \]

Conclusion

Thus, the correct response to the question about how many turning points Rogelio's sketch should include is:

exactly eight.