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.

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: