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, .

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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.

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Find the inflection point(s) of the function
Find the second derivative,
Where does change sign?
Is continuous there?
The function has an inflection point at

Find the inflection point(s) of the function
Find the second derivative,
Where does change sign?
Is continuous there?
The function has an inflection point at

Find the inflection point(s) of the function
Find the second derivative,
Where does change sign?
Is continuous there?

List multiple answers in ascending order.
The function has inflection points at and .

Find the inflection point(s) of the function
Find the second derivative,
Where does change sign?
Is continuous there?
If there are no inflection points, type “none”.
The function has inflection point(s) at

Find the inflection point(s) of the function
Find the second derivative,
Use the product rule to compute the derivatives
and is similar
Where does change sign?
Is continuous there?
The function has an inflection point at

Find the inflection point(s) of the function
Find the second derivative,
Use the product rule to compute the derivatives
and is similar
Where does change sign?
Is continuous there?
List multiple answers in ascending order.
The function has inflection points at and

Find the inflection point(s) of the function in the interval
Find the second derivative,
Where does change sign?
Is continuous there?
List multiple answers in ascending order.
On the interval , the function has inflection points at and .

Find the inflection point(s) of the function
Find the second derivative,
Use the Quotient Rule to compute the derivatives
Where does ?
To solve the equation ,
let and solve for first
Is continuous there?
List multiple answers in ascending order.
The function has inflection points at and .

Find the inflection point(s) of the function on the interval .
Find the second derivative,
The derivative of is
Use the Chain Rule to find the derivative of
Where does ?
On the interval , the function
has an inflection point at .

Here is a detailed, lecture style video on concavity:
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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.

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Use the Second derivative Test to find the local extremes of The critical numbers are (list in ascending order) and .
has a local maximum at and a local minimum at