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
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.
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 analyze the polynomial factors.
Let's start by looking at each factor separately:
1. The factor x^3 has 3 roots at x = 0 (multiplicity 3). It looks like a cubic function and can have at most 2 turning points.
2. The factor (x-5)^2 has 2 roots at x = 5 (multiplicity 2). It is a quadratic function and can have at most 1 turning point.
3. The factor (5-x)^4 has 4 roots at x = 5 (multiplicity 4). It is also a quadratic function and can have at most 1 turning point.
Now, considering these individual factors, we can determine the total number of turning points:
- The factor x^3 can have at most 2 turning points.
- The factor (x-5)^2 can have at most 1 turning point.
- The factor (5-x)^4 can have at most 1 turning point.
So, the maximum number of turning points on the graph of g(x) is 2 + 1 + 1 = 4. However, since the multiplicity of some roots is greater than 1, it is possible for some of the turning points to coincide.
Therefore, the correct answer is option 3: eight at most.
To determine the number of turning points in the graph of the function g(x) = 3x^3(x-5)^2(5-x)^4, we need to analyze the behavior of the function.
A turning point is a point on the graph where the function changes from increasing to decreasing or vice versa. It occurs when the derivative of the function changes sign.
To find the turning points, we first need to find the derivative of g(x). Let's denote g'(x) as the derivative of g(x):
g'(x) = d/dx [3x^3(x-5)^2(5-x)^4]
To find the derivative of the given function g(x), we can apply the product rule and chain rule.
Now, let me explain the steps to find the derivative of g(x):
1. Apply the product rule:
The product rule states that if we have two functions, f(x) and g(x), then the derivative of their product h(x) = f(x) * g(x) is given by:
h'(x) = f'(x) * g(x) + f(x) * g'(x)
Applying the product rule to g(x), we get:
g'(x) = 3x^2 * (x-5)^2 * (5-x)^4 + 3x^3 * 2(x-5) * (5-x)^4 + 3x^3 * (x-5)^2 * 4(5-x)^3
2. Expand and simplify the equation:
Multiply out the terms and simplify the expression obtained in step 1. This will give you the simplified form of g'(x).
3. Set the derivative equal to zero and solve for x:
To find the turning points, we need to find the values of x where the derivative g'(x) equals zero. Set g'(x) = 0 and solve for x.
4. Count the number of turning points:
The number of turning points is equal to the number of distinct solutions obtained in step 3.
By following these steps, Rogelio will be able to find the number of turning points in the graph of g(x).