To determine the number of turning points for the function \( g(x) = 3x^3(x - 5)^2(5 - x)^4 \), we can analyze the components of the function.
First, we identify the degree of the polynomial. The function can be expanded as follows:
- The term \( 3x^3 \) contributes a degree of 3.
- The term \( (x - 5)^2 \) contributes a degree of 2.
- The term \( (5 - x)^4 \) contributes a degree of 4.
So, the total degree of the polynomial \( g(x) \) is: \[ 3 + 2 + 4 = 9 \]
The maximum number of turning points for a polynomial is given by the degree minus one. Thus, for a polynomial of degree 9, the maximum number of turning points is \( 9 - 1 = 8 \).
Let’s also consider the nature of the roots based on their multiplicities in the function:
- The root \( x = 0 \) has a multiplicity of 3 (which means it can contribute 2 turning points).
- The root \( x = 5 \) has a multiplicity of 2 (which contributes 1 turning point).
- The root \( x = 5 \) also has a contribution in the form of \( (5 - x)^4 \) (which can contribute 3 turning points).
Checking the contributions:
- At the root \( x = 0 \), since it has odd multiplicity (3), it contributes 2 turning points.
- At the root \( x = 5 \), the even multiplicity (2) means it does not add additional turning points because it does not change direction but returns.
- At the root \( x = 5 \) from \( (5 - x)^4 \), it contributes 3 turning points because it has an even higher multiplicity.
Putting this together, maximum potential contributions suggest a possibility of unique turning points but usually conforms closer to the maximum of 8.
Considering that the maximum number of turning points derived earlier from the function's degree is 8, this is consistent with our assessment.
Therefore, the correct response is:
eight at most
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