Concavity

One of the applications of differentiation is to know whether a curve is concave upward or downward in a certain interval. We begin by putting a definition of concavity.

Definition

If the curve of the function f always remains above the tangent lines for every point in the interval I, we say that the curve is concave upward on that interval.

If the curve always remains below its tangent lines on I, we say it is concave downward on that interval.

By means of the second derivative, the next theorem helps us determine whether a function is concave upward or downward on an interval.

Theorem

Let f be a function defined on an interval I.

1.     The function is concave upward on I if  for all x inside I.

2.     The function is concave downward on I if  for all x inside I.

Note that this test of concavity tells us nothing when  or if it does not exist. There the function can be concave upward, downward or neither. The three cases are explained here by the graphs of the functions ,  and . The second derivative of the three functions is zero when x=0, but the point of inflection appears only in .

In such cases, the concavity of the function is determined by its concavity before and after that point. That will be shown in the examples.

Definition

When the concavity of the function changes after and before the point, it is a point of inflection. The second derivative at all points of inflection will be zero or it will not exist. At these points of inflection, the curve is neither concave upward or downward.

Example

Show how the concavity of the function  changes. Find the points of inflection.

Solution