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

Question

BlackTom AI · Accepted Answer

Let's examine the given function graph on the interval [a, b] and focus on the local maximum labeled on the left side of the graph.

Option: The graph of the function is concave down at its local maximum.

First, recall concavity intuition: a concave down region resembles a cap or a hill, where the slope is decreasing as x increases. At a local maximum, the function changes from increasing to decreasing, and if the section near the maximum curves downward (like an upside-down cup), that indicates concavity downward (second derivative negative) at the peak.

Looking at the plotted curve near the local maximum, the curve bends downward as it reaches the top, forming a hill-like shape. This indicates that the second derivative is negative around that point, meaning the graph is concave down at the local maximum.

If we consider potential alternatives (even though they are not listed as separate options here), a claim that the graph is concave up at the local maximum would require the curve to bend upward toward the peak, which is not what the picture shows. Similarly, a statement about concavity elsewhere (e.g., far to the left or right) would depend on the curvature in those regions, not at the local maximum itself.

In summary, based on the visual curvature at the local maximum, the graph is concave down at that point, supporting the given statement's truth value in this context.

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

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