Unit 3

Curve Sketching

Lesson 4: Concavity and Points of Inflection

So far, the focus of this Unit has been on the relationship between a function and its derivative. In this Lesson, attention will be given to the relationship between a function and its second derivative.

Interactive

Interactive

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Click the interactive button to open the applet Where is a Second Derivative Positive?. The applet shows a polynomial function that can be transformed by moving the points on the graph.

• Predict where the second derivative of this function will be positive.
• Use the checkboxes to check your prediction.
• Move some points on the graph of the original function and try again.

What characteristic(s) on the graph of the original function allow you to determine where the second derivative of that function will be positive or negative?