To determine the number of turning points on the graph of g(x), we need to analyze the behavior of the polynomial of g(x).
First, let's simplify the polynomial:
g(x) = 3x^3(x−5)^2(5−x)^4
Breaking it down, we have:
g(x) = 3x^3(x−5)(5−x)(x−5)(5−x)(x−5)(5−x)
g(x) = 3x^3(x−5)(5−x)(x−5)^2(5−x)^2
To find the number of turning points, we need to look at the exponents of the factors in the expression.
The exponent of (5-x) is 4, which means that the graph will change direction at this factor 4 times. So, on this factor alone, there are 4 turning points.
The exponent of (x-5) is 2, which means that the graph will change direction at this factor 2 times. So, on this factor alone, there are 2 turning points.
Since each factor contributes a unique turning point, the total number of turning points will be the sum of the turning points from each factor.
Therefore, the number of turning points on the graph of g(x) is: 4 + 2 = 6.
Thus, the answer is (4) nine at most.
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)
1. three at most
2. exactly eight
3. eight at most
4. nine at most
1 answer
1 answer