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: $\left(-\infty,0\right)$ , $\left(1,\infty\right)$ ; Concave down: $\left(0,1\right)$ ; Inflection points: $\left(0,0\right)$ , $\left(1,1\right)$

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

Concave down: $\left(-1,0\right)$ , $\left(1,\infty\right)$ ; Concave up: $\left(0,1\right)$ ; Inflection points: $\left(0,0\right)$ , $\left(1,1\right)$

Concave up: $\left(-1,0\right)$ , $\left(1,\infty\right)$ ; Concave down: $\left(0,1\right)$ ; Inflection points: $\left(-1,1\right)$ , $\left(0,0\right)$ , $\left(1,1\right)$

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. The second derivative tells us about the concavity of the function.

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 sign.

From the graph, observe the intervals where the curve is bending upwards (concave up) and downwards (concave down). The curve is concave up when it is shaped like a cup (U) and concave down when it is shaped like a cap (∩).

In the graph, the function appears to be concave down on the interval (0, 1) and concave up on the intervals (-∞, 0) and (1, ∞).

The inflection points, where the concavity changes, are at the points (0, 0) and (1, 1) as these are the points where the curve changes from concave down to concave up or vice versa.

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