Concavity and Points of Inflection
While the tangent line is a very useful tool, when it comes to investigate the graph of a
function, the tangent line fails to say anything about how the graph of a function "bends"
at a point. This is where the second derivative comes into play.
Example. Consider the function
f(x) = ax^{2}. The tangent
line at 0 is the xaxis regardless of the value of a. But if
we change a, the graph of f(x) bends more or less sharply
depending on the size of the parameter a:
Note that the value a is directly related to the second
derivative, since
f ''(x) = 2a.
Definition. Let f(x) be a differentiable function on an
interval I.
 (i)
 We will say that the graph of f(x) is concave up on I iff f '(x) is increasing on I.
 (ii)
 We will say that the graph of f(x) is concave down on I iff f '(x) is decreasing on I.
Some authors use concave for concave down and
convex for concave up instead.
Usually graphs have regions which are concave up and others which
are concave down. Thus there are often points at which the graph
changes from being concave up to concave down, or vice versa.
These points are called inflection points. Since the
monotonicity behavior of a function is related to the sign of its
derivative we get the following result:
Let f(x) be a differentiable function on an interval I.
Assume that f '(x) is also differentiable on I.
 (i)
 f(x) is concave up on I iff
on I.
 (ii)
 f(x) is concave down on I iff
on I.

It is clear from this result that if c is an inflection point
then we must have
Example. Consider the function
f(x) = x^{9/5}  x. This
function is continuous and differentiable for all x. We have
Clearly f ''(0) does not exist. In fact, f '(x) has a vertical
tangent at 0. More precisely we have for
which implies
 (1)

f ''(x) > 0 for x > 0;
 (2)

f ''(x) < 0 for x < 0.
In other words, the point 0 is a point of inflection even though
f ''(0) does not exist.
Remark. Note that if the graph is concave up (resp.
concave down), then the tangent line at any point is below (resp.
above) the graph. Therefore, at an inflection point the graph
"cuts" through the tangent line.
Exercise 1. Describe the concavity of the graph of
for
.
Answer.
Exercise 2. Find
and
so that the function
has a point of inflection at
.
Answer.
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