We were lead to think about the second derivative by studying the
behaviour of a function near its critical points. We can summarize
that discussion by stating:
 If
is a critical point and
, then is a local
maximum.
 If
is a critical point and
, then is a local
minimum.
We can understand this more fully by thinking geometrically about
the behaviour of the tangent lines. Let suppose that
in some region. This means that the
derivative
, or the slope of the tangent lines,
is increasing in this region. Here is a possible picture of the
tangent lines.
If we now imagine the corresponding shape for the graph of the
original function, it must be something like this:
We say that a graph which is curved in this way is concave up
. The second derivative being positive implies that the graph is
concave up since the slope of tangent lines must be increasing. Here
is another example of a graph which is concave up.
Conversely, when the second derivative is negative, the slope of
the tangent lines is decreasing. Here is a possible picture of the
tangent lines:
If we now imagine the corresponding shape for the graph of the
original function, it must be something like this:
We say that a graph is concave down
when it is curved in this way. When the second derivative is
negative, it implies that the graph is concave down
since the slope of tangent lines must be decreasing. Here
is another example of a graph which is concave down.
An important point on a graph is one which marks a transition
between a region where the graph is concave up and one where it is
concave down. We call such a point an inflection point .
Have a look again at the graph of the function
and see if you can find an inflection point. How does the
behaviour of the tangent lines change at this point?
