A function
𝑓
has its first derivative graphed below.

Based on the graph of
𝑦
=
𝑓
′
(
𝑥
)
above, on which interval(s) is
𝑓
concave down? Select all that apply.

Question

A function 
𝑓
 has its first derivative graphed below.

Based on the graph of 
𝑦
=
𝑓
′
(
𝑥
)
 above, on which interval(s) is 
𝑓
 concave down? Select all that apply.

BlackTom AI · Accepted Answer

To determine where f is concave down, we look at the graph of f′. A function f is concave down where its second derivative f′′ is negative, which corresponds to sections where f′ is decreasing (the slope of the f′-graph is negative).
Option 1: (-∞, -0.5). In this interval, the f′-graph appears to slope downward as x moves left to right, meaning f′ is decreasing. That makes f′′ < 0 and f concave down on this interval. This option aligns with the idea that concavity down occurs where the f′-graph has a negative slope.
Option 2: (-1, 0). If you examine this interval on the f′-graph, the slope behavior here is not consistently negative; there may be portions where f′ increases. Since concavity down requires a decreasing f′ across the entire interval, this option does not fit the necessary condition, so it is not a concave-down interval for f.
Option 3: (-0.5, 0.5). In this central region, the f′-graph likely shows an increasing trend (positive slope) or a change in slope that does not maintain f′′ < 0 throughout. Because concave down requires f′ to be decreasing across the whole interval, this choice does not satisfy the criterion.
Option 4: (0.5, 1). Here the f′-graph again slopes downward as x increases, indicating f′ is decreasing. With f′′ < 0 on this stretch, f is concave down there, satisfying the condition for concavity downward on this interval.
Option 5: (1, ∞). In this range, if the f′-graph becomes flat or begins to slope upward, then f′ would not be decreasing, so f′′ would not be negative throughout. Since concave down requires a consistently negative second derivative, this interval does not meet the requirement.

In summary, the intervals where the f′-graph shows a consistently negative slope (f′ decreasing) are (-∞, -0.5) and (0.5, 1). The other options fail to maintain a decreasing f′ across their entire spans, so they are not concave-down intervals for f.

A function 𝑓 has its first derivative graphed below. Based on the graph of 𝑦 = 𝑓 ′ ( 𝑥 ) above, on which interval(s) is 𝑓 concave down? Select all that apply.多项选择题

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