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point on a graph where the concavity of the curve changes (from concave down to concave up, or vice versa) is called a point of inflection (Definition 4.14). By implication (think about what separates positive and negative numbers on a number line), if a point (c, f (c)) is a point of inflection, then f ′ ′ ( c ) = 0 .
This is a self contained set of lecture notes for Math 221. The notes were written by Sigurd Angenent, starting from an extensive collection of notes and problems compiled by Joel Robbin.
All points of inflection on the graph of must occur either where equals zero or where is undefined. In the following examples, we will use these properties to sketch graphs
Definition. An inflection point is a point (c, f (c)) on the graph of f where the concavity changes. At such a point, either f ′′(c) = 0 or f ′′(c) does not exist. Procedure for finding the Inflection Points. Step 1. Compute f ′′(x) and determine all points in the domain of f where either f ′′(c) = 0 or f ′′(c) does not exist. Step 2.
Definition — point of inflexion. A point of inflexion is where the curve changes from convex (concave-up) to concave (concave-down), or vice versa (as in the case of y = x3 ).
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Goal: Use Mathematica to identify inflection points. Task 1. By definition, f(x) has an inflection point at (a, f(a)) as long as three conditions are satisfied: f(x) is continuous at x=a. f’(x) does not change sign at x=a. f”(x) changes sign at x=a. We define a cubic function:
