Warm up

4 3 212 48 64y x x x x

1. Look at the intervals where the function is increasing

2. Try to describe what looks different about the regions where the function is increasing

Chapter Three: Section Four The word we will use to describe the different

behaviors of the increasing regions of the function is the word concavity.

When the second derivative of a function is positive, we say that the function is concave up. What this means physically is that the movement of the graph has positive acceleration.

When the second derivative is negative then the function is said to be concave down and this means that the acceleration of the graph is negative.

Concavity

First derivative:

y is positive Curve is rising.

y is negative Curve is falling.

y is zero Possible local maximum or minimum.

Second derivative:

y is positive Curve is concave up.

y is negative Curve is concave down.

y is zero Possible inflection point(where concavity changes).

Example Determining Concavity 2Use the Concavity Test to determine the concavity of ( ) on the

interval (2,8).

f x x

2Since " 2 is always positive, the graph of is concave

up on any interval. In particular, it is concave up on (2,8).

y y x

Inflection Point

1. Point where concavity changes 2. Point where second derivative changes

sign3. POSSIBLY happen when f “ = 0

or f ’’ is undefined

Find the points of inflection and discuss the concavity

34 22

1)( xxxf

Find the points of inflection and discuss the concavity

34 4)( xxxf

Ex01: Determining Concavity

3

6)(

2 x

xf

Ex02: Determining Concavity

4

1)(

2

2

x

xxf

Learning about Functions from Derivatives