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
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.
appears to be “curving downward” until x = 0.
appears to be “curving upward” from x = 0 onward.
has a relative maximum at (-1, 2) where is “curving downward” and a minimum at (1, -2) where is “curving upward.”
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:
is concave down when x < 0
(notice the second derivative is negative).
is concave up when 0 < x
(notice the second derivative is positive).
has a relative maximum at (-1, 2) and .
has a relative minimum at (1, -2) and .
Use the following information about
to answer the next three questions: