Let be the function defined on whose graph is given below, and let .

Which of the following statements is true?

Question

Let  be the function defined on  whose graph is given below, and let  .

Which of the following statements is true?

BlackTom AI · Accepted Answer

The problem presents a function F defined on some domain with a graph provided, and asks which statement is true. Unfortunately, the answer options are not shown in the data you provided, so we cannot evaluate each possible choice against the graph.

What we can do is discuss how you would verify the given statement, 'F has a local maximum at 4,' if the graph were visible:
- Look at the graph around x = 4. A local maximum occurs at x = 4 if the y-value reaches a peak relative to nearby points, meaning the function values to the left and right of x = 4 are less than F(4).
- On a smooth graph, you would also expect the derivative to change from positive to negative as you pass through x = 4 (though the derivative criterion is not strictly necessary for a graphical check, it aligns with the intuition of a peak).
- If the graph has a plateau at x = 4 where the function is constant over an interval containing 4, that would be a local maximum at each point of that interval only if that constant value is greater than neighboring values; otherwise it would not be a strict local maximum.
- If the graph shows the point (4, F(4)) rising to the left and then falling to the right, or if 4 corresponds to a labeled vertex where the curve attains its highest nearby value, that would support the statement.

Without the actual options, we cannot compare the claim to alternative statements (e.g., 'F has a local minimum at 4,' 'F has an inflection point at 4,' etc.). If you can share the missing answer choices or the graph, I can provide a precise option-by-option analysis including why the statement about a local maximum at 4 is correct or incorrect.

MATH_1225_64262_202506 5.3 The Fundamental Theorem of Calculus

Let be the function defined on whose graph is given below, and let . Which of the following statements is true?

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