5. Graphical Applications of Derivatives

Concavity

5. 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: $\left(-2,0\right)$ , $\left(2,4\right)$ ; 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

To determine where the function is concave up or down, we need to analyze the second derivative of the function. Concavity is determined by the sign of the second derivative: if it's positive, the function is concave up; if it's negative, the function is concave down.

Identify the points where the concavity changes, known as inflection points. These occur where the second derivative is zero or undefined, and the concavity changes from up to down or vice versa.

From the graph, observe the intervals where the curve is bending upwards (concave up) and downwards (concave down). Look for changes in the direction of bending to identify inflection points.

Based on the graph, the function appears to be concave up on the intervals (-2, 1) and (2, 4). This is where the curve is bending upwards.

The function is concave down on the intervals (1, 2) and (4, 5). Inflection points, where the concavity changes, are approximately at (1, 0), (2, -1), and (4, 3).

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