In this section we learn about the two types of curvature and determine the curvature of a function.
Concavity
In this section we will discuss the curvature of the graph of a given function. There are two types of curvature: concave up and concave down. The main tool for discussing curvature is the second derivative, .
The following theorem helps us to determine where a function is concave up and
where it is concave down.
The last example brings up a new concept. The function changes concavity at . We
call this an inflection point of the function.
In the next two examples, we will discuss the curvature of the given functions and find their inflection points.
The second derivative of is . To determine where is positive and where it is negative, we will first determine where it is zero. Hence, we will solve the equation for .
We have so . This value breaks the real number line into two intervals, and . The
second derivative maintains the same sign throughout each of these intervals. To
determine whether it is positive or negative, we choose a test point in each interval.
For the interval, , we choose . Plugging this into the second derivative, we get . Since
this value is negative, we can conclude that the original function, is concave down on
the interval .
Next, we choose the test point for the interval . Plugging this into the second derivative gives . Since this value is positive, we can conclude that the original function, is concave up on the interval . Finally, noting that the original polynomial is continuous everywhere and changes concavity at , we can say that has an inflection point at .
We have so . This value breaks the real number line into two intervals, and . The
second derivative maintains the same sign throughout each of these intervals. To
determine whether it is positive or negative, we choose a test point in each interval.
For the interval, , we choose . Plugging this into the second derivative, we get . Since
this value is negative, we can conclude that the original function, is concave down on
the interval .
Next, for the interval , we choose the test point . Plugging this into the second derivative gives . Since this value is positive, we can conclude that the original function, is concave up on the interval . Finally, noting that the original polynomial is continuous everywhere and changes concavity at , we can say that the original function, has an inflection point at .
List multiple answers in ascending order.
The function has inflection points at and .
The function has inflection point(s) at
These two values break the real number line into three intervals: and with test points and , respectively. Plugging these into the second derivative gives and Hence is concave up on the intervals and and it is concave down on the interval . Finally, we can infer from this that the continuous function has inflection points at
The function has inflection points at and
List multiple answers in ascending order.
On the interval , the function has inflection points at and .
These two values of break the real number line into three intervals: and . The sign of will remain the same on each of these intervals. To determine the sign on each interval, we use the test points and respectively. Plugging the test points into the second derivative gives and From this information we can conclude that the original function, is concave up on the intervals and and concave down on the interval . Finally, since the original function is continuous everywhere, we can say that are inflection points for .
The function has inflection points at and .
On the interval , the function
has an inflection point at .
We next explore a special relationship between concavity and local extremes.
The following figure should convince the reader of the validity of the Second Derivative Test.