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
exactly eight
exactly eight
eight at most
eight at most
nine at most
The correct answer is "eight at most." The graph of the function g(x) could have a maximum of eight turning points, but it could also have fewer.
The number of turning points in a graph is determined by the power of the polynomial function. In this case, the function g(x) is a polynomial of degree 3 multiplied by a polynomial of degree 2 multiplied by a polynomial of degree 4.
Since the highest degree term in g(x) is x^3, the maximum number of turning points for this function would be 3. Therefore, the correct answer is "three at most".
To determine the number of turning points on the graph of the function g(x) = 3x^3(x-5)^2(5-x)^4, we need to consider the behavior of the function as it changes direction.
A turning point occurs when the function changes from increasing to decreasing (concave down) or from decreasing to increasing (concave up). We can identify these turning points by finding the points where the first derivative is equal to zero or undefined.
To find the first derivative of g(x), we can use the product rule and the chain rule.
g'(x) = 3[3x^2(x-5)^2(5-x)^4 + x^3 * 2(5-x)^4 + x^3(x-5)^2 * 4(5-x)^3]
Next, we solve the equation g'(x) = 0 to find the x-values of the turning points.
Setting g'(x) = 0:
3[3x^2(x-5)^2(5-x)^4 + 2x^3(5-x)^4 + 4x^3(x-5)^2(5-x)^3] = 0
Now, we can solve this equation to find the x-values of the turning points. However, due to the complexity of the equation, it's difficult to determine the exact number of turning points without going through the calculation process.
So the answer is "nine at most" since it's possible that there could be nine turning points, although the exact number cannot be determined without solving the equation explicitly.