6. Graphical Applications of Derivatives

Concavity

6. Graphical Applications of Derivatives

Concavity

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Multiple Choice

For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

Concave down: $\left(0,\infty\right)$ ; No inflection pt

Concave down: $\left(0,\infty\right)$ ; Inflection pt: $\left(0,0\right)$

Concave up: $\left(-\infty,0\right)$ ; Concave down: $\left(0,\infty\right)$ ; Inflection pt: $\left(0,0\right)$

Concave down: $\left(-\infty,\infty\right)$ ; No inflection pt

Verified step by step guidance

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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