Let be a continuous function and let be in the domain of . Suppose that when and that when . Then is a point of inflection .

Question

Let  be a continuous function and let  be in the domain of . Suppose that   when  and that  when  . Then  is a point of inflection .

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The question presents a continuous function and indicates some conditions at certain points, followed by a statement about an inflection point, but the exact mathematical expressions are missing in the prompt. To analyze a multiple-choice (or dropdown) item properly, we would need the explicit answer options to evaluate each one against the given information.

Given what's provided: we know the function is continuous on its domain. Inflection points are points where the concavity changes, which typically relates to the second derivative (if it exists) or to a sign change in the second derivative (or a change in concavity observed from a graph). Without the actual formulas or conditions (for example, the behavior of f''(x) or f'(x) around the candidate point), we cannot deterministically conclude whether the candidate point is indeed a point of inflection.

Also, the prompt mentions that the answer field lists "point of inflection" as the correct option, but without the surrounding answer choices, we cannot verify the correctness or provide reasoning specific to each option.

In a typical setup, these are the kinds of considerations that would guide evaluation:
- If the second derivative f''(x) exists near the point and changes sign, that point is a point of inflection.
- If the second derivative exists and is zero at the point but does not change sign, it is not an inflection point.
- If the concavity changes due to higher-order behavior or if the second derivative does not exist but the concavity changes, the point can still be an inflection point in a broader sense.
- If the provided conditions at other points imply monotonic or convex/concave behavior that contradicts a change in concavity at the candidate point, that would disallow it.

Because the actual answer options are missing, we cannot map each option to a true/false assessment or explain why one option is correct and others are not. If you provide the list of answer options (the dropdown choices) or the explicit expressions for the conditions around the candidate point, I can produce a detailed, option-by-option analysis explaining why each choice is correct or incorrect, with varied sentence structures and a clear, readable step-by-step rationale.

MATH_1225_17255_202501 4.3 How Derivative Affect the Shape of a Graph

Let be a continuous function and let be in the domain of . Suppose that when and that when . Then is a point of inflection .

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