Rogelio is asked to sketch a graph of g(x)=3x^3(x−5)^2(5−x)^4. How many turning points should his sketch include?(1 point)
three at most
exactly eight
nine at most
eight at most
eight at most
To determine the number of turning points on the graph of g(x) = 3x^3(x−5)^2(5−x)^4, we need to examine the behavior of the function.
The turning points occur where the graph changes from increasing to decreasing, or vice versa. These points can be found by analyzing the sign of the first derivative.
We start by finding the first derivative of g(x):
g'(x) = 3x^3(x−5)^2(5−x)^4'
To determine the sign of g'(x) and the number of turning points, we need to determine the sign of each term within g'(x). Let's analyze the sign of each factor.
1) 3x^3 - This term represents a cubic function. Since the coefficient (3) is positive, the overall sign of this term will depend on the sign of x. So, we have one turning point for this term.
2) (x−5)^2 - This term represents a quadratic function. The squared term ensures that it is always positive. Therefore, it does not contribute to any turning points.
3) (5−x)^4 - This term represents a quartic function. Again, the exponent is even, so this term does not change sign. Therefore, it does not contribute to any turning points.
By combining all three factors, we find that there is only one turning point on the graph of g(x).
Therefore, the answer is "at most one turning point," which is not listed in the given answer choices.