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 up: $\left(-2,-1\right)$ , $\left(2,4\right)$ ; Concave down: $\left(-1,1\right)$ , $\left(1,2\right)$ , $\left(4,5\right)$ ; Inflection points: $\left(-1,0\right)$ , $\left(1,0\right)$ , $\left(2,-1\right)$ , $\left(4,3\right)$

Concave up: $\left(-2,-1\right)$ , $\left(2,4\right)$ ; Concave down: $\left(-1,2\right)$ , $\left(4,5\right)$ ; Inflection points: $\left(0,1\right)$ , $\left(1,0\right)$ , $\left(2,-1\right)$ , $\left(4,3\right)$

Concave up: $\left(-2,1\right)$ , $\left(2,4\right)$ ; Concave down: $\left(1,2\right)$ , $\left(4,5\right)$ ; Inflection points: $\left(1,0\right)$ , $\left(2,-1\right)$ , $\left(4,3\right)$

Concave up: $\left(-2,0\right)$ $\left(-2,1\right)$ , $\left(2,4\right)$ ; Concave down: $\left(0,2\right)$ , $\left(4,5\right)$ ; Inflection points: $\left(0,1\right)$ , $\left(1,0\right)$ , $\left(2,-1\right)$ , $\left(4,3\right)$

Verified step by step guidance

Step 1: Understand the concept of concavity. A function is concave up on an interval if its second derivative is positive on that interval, and concave down if its second derivative is negative. Inflection points occur where the concavity changes, i.e., where the second derivative changes sign.

Step 2: Analyze the graph visually. Look for regions where the curve bends upwards (concave up) or downwards (concave down). Identify points where the curve transitions between these behaviors, as these are potential inflection points.

Step 3: From the graph, observe the intervals of concavity. The function appears concave up on the intervals (-2, 1) and (2, 4), as the curve bends upwards. It appears concave down on the intervals (1, 2) and (4, 5), as the curve bends downwards.

Step 4: Identify the inflection points. These are the points where the curve changes from concave up to concave down or vice versa. From the graph, the inflection points are approximately at (1, 0), (2, -1), and (4, 3).

Step 5: Summarize the findings. The function is concave up on (-2, 1) and (2, 4), concave down on (1, 2) and (4, 5), and has inflection points at (1, 0), (2, -1), and (4, 3).

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