객관식
For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.
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6. Graphical Applications of Derivatives
Concavity
객관식
For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.
A
Concave down: ; No inflection pt
B
Concave down: ; Inflection pt:
C
Concave up: ; Concave down: ; Inflection pt:
D
Concave down: ; No inflection pt
검증된 단계별 안내1
Step 1: Observe the graph and identify the behavior of the curve. The graph shows a function that starts steeply decreasing and then transitions to a gentler slope as x increases.
Step 2: Recall that concavity is determined by the second derivative of the function. If the second derivative is positive, the function is concave up; if negative, the function is concave down.
Step 3: Analyze the graph visually. From x = -∞ to x = 0, the curve appears to be concave up because the slope is increasing (becoming less steep). From x = 0 to x = ∞, the curve appears to be concave down because the slope is decreasing (becoming flatter).
Step 4: Identify the inflection point. An inflection point occurs where the concavity changes, which is at x = 0 in this graph. At this point, the function transitions from concave up to concave down.
Step 5: Summarize the intervals of concavity and the inflection point. The function is concave up on (-∞, 0), concave down on (0, ∞), and has an inflection point at (0, 0).
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