Question
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The graph of a rational function has a local minimum at left parenthesis 7 comma 0 right parenthesis. The complex number 4 + 2i is a zero of the function. What is the least possible degree of the function?
(1 point)
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The graph of a rational function has a local minimum at left parenthesis 7 comma 0 right parenthesis. The complex number 4 + 2i is a zero of the function. What is the least possible degree of the function?
(1 point)
Responses
2
2
3
3
4
4
5
5
Answers
GPT-4o mini
To determine the least possible degree of the rational function given the information, let's analyze the components:
1. **Local Minimum at (7, 0)**: A local minimum implies that the function changes direction here, suggesting that there is at least one factor (x - 7) in the function. Moreover, since it's a local minimum and located on the x-axis, the factored form would indicate that (x - 7) is raised to at least an even power (to ensure a minimum).
2. **Complex Zero 4 + 2i**: For a rational function, complex zeros occur in conjugate pairs. This means that if 4 + 2i is a zero, then its conjugate, 4 - 2i, must also be a zero. Therefore, we have two distinct linear factors: (x - (4 + 2i))(x - (4 - 2i)), which will contribute a total degree of 2.
Putting this together:
- The factor related to the local minimum at x = 7 suggests a minimum condition, implying at least one (x - 7) factor.
- Since we need (x - 7) to be raised to an even power for a minimum, this contributes at least 2 to the degree.
- The complex roots contribute another 2 to the degree.
So, we have a minimum degree of \(2 \, (from \, the \, minimum) + 2 \, (from \, the \, conjugate \, zeros) = 4\).
Thus, the least possible degree of the function is **4**.
So the answer is:
4
1. **Local Minimum at (7, 0)**: A local minimum implies that the function changes direction here, suggesting that there is at least one factor (x - 7) in the function. Moreover, since it's a local minimum and located on the x-axis, the factored form would indicate that (x - 7) is raised to at least an even power (to ensure a minimum).
2. **Complex Zero 4 + 2i**: For a rational function, complex zeros occur in conjugate pairs. This means that if 4 + 2i is a zero, then its conjugate, 4 - 2i, must also be a zero. Therefore, we have two distinct linear factors: (x - (4 + 2i))(x - (4 - 2i)), which will contribute a total degree of 2.
Putting this together:
- The factor related to the local minimum at x = 7 suggests a minimum condition, implying at least one (x - 7) factor.
- Since we need (x - 7) to be raised to an even power for a minimum, this contributes at least 2 to the degree.
- The complex roots contribute another 2 to the degree.
So, we have a minimum degree of \(2 \, (from \, the \, minimum) + 2 \, (from \, the \, conjugate \, zeros) = 4\).
Thus, the least possible degree of the function is **4**.
So the answer is:
4