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: $\left(-\infty,0\right)$ , $\left(1,\infty\right)$ ; 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

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

Step 2: Analyze the graph provided. Observe the curvature of the graph. The graph is concave down when it curves downward like a frown and concave up when it curves upward like a smile.

Step 3: Identify the intervals of concavity. From the graph, the function appears concave down on the interval (0,1) because the graph curves downward. It is concave up on the intervals (−∞,0) and (1,∞) because the graph curves upward.

Step 4: Locate the inflection points. Inflection points occur where the graph changes concavity. From the graph, the concavity changes at (0,0) and (1,1), making these the inflection points.

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

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