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?
Responses
three at most
three at most - no response given
exactly eight
exactly eight - incorrect
eight at most
eight at most - not selected, this is the correct answer
nine at most
explain how you get that answer
To determine the number of turning points in the graph of g(x), we need to consider the factors of the function.
First, let's look at the factor (x−5)^2. This factor represents a parabola with a turning point at x = 5. Therefore, there is one turning point associated with this factor.
Next, let's consider the factor (5−x)^4. This factor also represents a parabola with a turning point at x = 5. However, since the exponent is 4, the parabola is flattened and only touches the x-axis at the turning point. Therefore, there are no additional turning points associated with this factor.
Finally, let's analyze the factor 3x^3. This factor is a cubic function and can have up to two turning points.
Since these factors are multiplied together, the total number of turning points in the graph of g(x) is the sum of the turning points from each factor, which is:
1 (from the factor (x−5)^2) + 0 (from the factor (5−x)^4) + 2 (from the factor 3x^3) = 3.
Therefore, Rogelio's sketch of g(x) should include three turning points at most.