 Site Navigation                          Introductory Calculus: Second Derivative Test 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:    Examples:   Put the information about the first and second derivatives into sign charts.    Sketch a possible graph of paying close attention to the points given, the regions of increase and decrease, and the regions of differing concavity.    At what value(s) of is there a relative min or max? How do you know?

Use the following information about to answer the next two questions.    Examples:   Find the critical points of from the first derivative. Use these to construct a sign chart for .    Find the critical points of from the second derivative. Use these to construct a sign chart for . M Ransom

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