Regions of Concavity up or down AND Maximum and Minimum Values of a Function

We look first at an example function . We calculate the derivative:

 

 

Notice that the derivative equals 0 when x = -1 or +1.

 

It can be shown that the derivative is positive when x < -1 and x > +1, and is negative when -1 < x < +1.

 

Next we calculate the second derivative:

 

.

 

Notice that the second derivative is 0 when x = 0.

 

It can be shown that the second derivative is negative for x < 0 and positive for x > 0.

 

Below, we show the graph of  twice. In the second graph, tangent lines are drawn which are horizontal (reminder: lines which are horizontal have a slope of 0). Some observations about these diagrams are made below the graphs.

 

 

1. appears to be “curving downward” until x = 0.
2. appears to be “curving upward” from x = 0 onward.
3. has a relative maximum at (-1, 2) where is “curving downward” and a minimum at (1, -2) where is “curving upward.”
4. changes from concave down to concave up at the point (0, 0) and the second derivative is 0.

 

These observations can be summarized as follows:

 

1. is concave down when x < 0
        (notice the second derivative is negative).
2. is concave up when 0 < x
        (notice the second derivative is positive).
3. has a relative maximum at (-1, 2) and .
4. has a relative minimum at (1, -2) and .
    

Generalized Results

A function is concave up at a point if there is an interval containing c and the second derivative is positive on this interval. If , there is a relative minimum at . A function is concave down at a point if there is an interval containing c and the second derivative is negative on this interval. If , there is a relative maximum at . A function possibly has a point of inflection at a point where the second derivative is exactly 0 (or as we shall see, at a point where the second derivative does not exist).

Use the following information about  to answer the next three questions:

Use the following information about  to answer the next two questions.