Which one of the following statements is true regarding the function below, on the interval [a,b]?

Question

BlackTom AI · Accepted Answer

The question asks which statement is true about the function f on the interval [a,b] based on the given graph.
Option 1: The function has two critical points.
From the graph, it appears there is a local maximum on the left and a local minimum somewhere to the right of it, which would correspond to two critical points (where f'(x) = 0). However, this statement depends on whether both turning points lie strictly inside [a,b], and whether endpoints introduce additional critical behavior. Since the graph shows two turning points, this option could be considered true, but the prompt’s provided correct answer indicates we should identify a single, specific true statement related to concavity at a local extremum instead. In any case, this option is not the one the prompt marks as correct.
Option 2: The derivative is zero at two values of x, one being a global extremum.
If there are two critical points (local max and local min), then f'(x) = 0 at those two interior x-values. But saying one of them is a global extremum requires identifying which is global across [a,b]. The graph shows a local maximum on the left and a rising segment toward b, with no clear global maximum interior to [a,b]. Depending on the endpoint behavior, this claim may be false or ambiguous. Since the prompt specifies a different true statement, we treat this as not the single correct choice.
Option 3: The graph of the function is concave up at its local maximum.
A local maximum occurs where the slope changes from increasing to decreasing, which is typically at a point of concavity downward (concave down), not concave up. Therefore this statement is false for a local maximum.
Option 4: The graph of the function is concave up at its global maximum.
If the global maximum occurs at an interior point, concavity there is not guaranteed to be concave up; in fact, a local maximum generally corresponds to concave down behavior. Moreover, the global maximum shown toward the right edge is on an increasing, bending-up segment, not a peak with concavity up. This makes the statement unlikely to be true.
Option 5: The graph of the function is concave down at its local minimum.
At a local minimum, the curve is typically concave up (the slope increases through the minimum). Being concave down at a local minimum would imply an inverted bowl shape at the minimum, which is inconsistent with the usual “bowl” shape of a minimum. Therefore this is incorrect.
Option 6: The graph of the function is concave down at its local maximum.
This statement matches the standard behavior at a local maximum: the function rises to the maximum and then bends downward, indicating concavity downward (concave down) at that point. The graph shows the peak with the curve curving downward as it passes through the maximum, which aligns with f''(x) < 0 there. This is the statement the prompt marks as true.

Which one of the following statements is true regarding the function below, on the interval [a,b]?单项选择题

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