Concavity and Inflection Points

In this topic:

    1. Define First Order and Second Order Critical Points
    
    2. Define Concave Up and Concave Down
    
    3. Define Inflection Point
    
    4. State the Concavity Test
    
    5. Illustrate with the functions and
    

    A portion of a graph that is cupped upward is called concave up and a portion of that is cupped downward is called concave down. The slope of a graph increases on an interval where the graph is concave up and decreases where the graph is concave down.

Definition (Critical Points) We will call the number a first order critical number if or does not exist and a second-order critical number if or does not exist.

Definition (Concave Up and Concave Down) If the graph of lies above all of its tangents on an interval it is called concave upward on If the graph of lies below all of these tangents, it is called concave downward on

Definition (Infection Point) A point on a curve is called an inflection point of the graph is concave up on one side of and concave down on the other side.

Proposition (Test for Concavity) Suppose is twice differentiable on an interval Then,

    (i) If for all in then the graph of is concave upward on

    (ii) If for all in then the graph of is concave downward on
    

Example (Concavity and Inflection Points) Determine where the curve is concave upward, where it is concave downward, and where the points of inflection are (if any).

    Solution. We use the Concavity Test and find the first and second derivatives:
    

    

    
which allows us to find the second order critical numbers, namely when and We summarize the Concavity Test in the following table:  



Therefore, is concave up on and concave down on The points and are inflection points.

Example (Concavity and Inflection Points) Determine where the curve is concave upward and where it is concave downward. Find all inflection points, local extrema, and sketch the curve.

    Solution. We apply the first derivative test and the concavity test by find the first and second derivatives





and then finding the first and second order critical numbers, namely:





Therefore, the first order critical numbers are and the second order critical numbers are We summarize the First Derivative Test and the Concavity Test in the following table:



Therefore, the function has a local maximum at and a local minimum at This function is increasing on the and decreasing on The point is an inflection point because is concave up on the interval and concave down on

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