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Question

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

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Answer 1

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:

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