MAT 1221Survey of Calculus

Section 3.3Concavity and the Second

Derivative Test

http://myhome.spu.edu/lauw

Expectations Check your algebra. Check your calculator works Formally answer the question with the

expected information

1 Minute… You can learn all the important concepts

in 1 minute.

1 Minute… Critical numbers – give the potential local

max/mins

1 Minute… Critical numbers – give the potential local

max/mins

If the graph is “concave down” at a critical number, it has a local max

1 Minute… Critical numbers – give the potential local

max/mins

If the graph is “concave up” at a critical number, it has a local min

1 Minute… You can learn all the important concepts

in 1 minute. We are going to develop the theory

carefully so that it works for all the functions that we are interested in.

There are a few definitions…

Preview Define

• Second Derivative• Concavities

Find the intervals of concave up and concave down

The Second Derivative Test

Second Derivative

Second Derivative 5 32f x x x

ddx

4 25 6f x x x

320 12f x x x

ddx

Given a function

which is a function.

)( of derivativefirst the)( of derivative the)(xf

xfxf

Higher Derivatives

Given a function

)( of derivative second the)( of derivative the

)()(

xfxf

xfdxdxf

Higher Derivatives

Concave Up(a) A function is called concave upward

on an interval if the graph of lies above all of its tangents on .

(b) A function is called concave downward on an interval if the graph of lies below all of its tangents on .

Concavity is concave up on

Potential local min.

Concavity is concave down on

Potential local max.

Concavity

has no local max. or min. has an inflection point at

yConcave

down

Concave up

xc

Definition An inflection point is a point where the

concavity changes (from up to down or from down to up)

Concavity Test

(a) If on an interval , then is concave upward on .(b) If on an interval , then f is concave downward on .

Concavity Test

(a) If on an interval , then is concave upward on .(b) If on an interval , then f is concave downward on .

Why? (Hint: ) ( ) ( )df x f xdx

Why? implies is increasing. i.e. the slope of tangent lines is increasing.

( ) ( )df x f xdx

Why? implies is decreasing. i.e. the slope of tangent lines is decreasing.

( ) ( )df x f xdx

Example 1Find the intervals of concavity and the inflection points

1362)( 23 xxxxf

Example 11362)( 23 xxxxf

1. Find , and the values of such that

)(xf )(xf

x 0)( xf

Example 11362)( 23 xxxxf

2. Sketch a diagram of the subintervals formed by the values found in step 1. Make sure you label the subintervals.

Example 11362)( 23 xxxxf

3. Find the intervals of concavity and inflection point.

1 8f

Example 11362)( 23 xxxxf

The Second Derivative TestSuppose is continuous near .(a) If and , then has a local minimum at c.(b) If and , then f has a local maximum at .

(c) If , then no conclusion (use 1st derivative test)

Second Derivative TestSupposeIf then has a local min at

0)( cf0)( cf

c

𝑓 ”(𝑐)>0

𝑓 ’ (𝑐)=0

x

y

Second Derivative TestSupposeIf then has a local max at 0)( cf

0)( cf

c

𝑓 ”(𝑐)<0

𝑓 ’ (𝑐)=0

x

y

The Second Derivative Test(c) If , then no conclusion

The Second Derivative TestIf , then no conclusion

4

3

2

2

( )

( ) 4 0 0

( ) 12

(0) 12 0 0

f x x

f x xx

f x x

f

The Second Derivative TestIf , then no conclusion

4

3

2

2

( )

( ) 4 0 0

( ) 12

(0) 12 0 0

g x x

g x xx

g x x

g

The Second Derivative TestIf , then no conclusion

3

2

( )

( ) 3 0 0

( ) 6(0) 6 0 0

h x x

h x xx

h x xh

The Second Derivative TestSuppose is continuous near .(a) If and , then has a local minimum at c.(b) If and , then f has a local maximum at .

(c) If , then no conclusion (use 1st derivative test)

Example 2Use the second derivative test to find the local max. and local min.

10249)( 23 xxxxf

Example 2(a) Find the critical numbers of

10249)( 23 xxxxf

Example 2(b) Use the Second Derivative Test to find the local max/min of

10249)( 23 xxxxf

The local max. value of isThe local min. value of is

2 10, 4 6f f

Review Example 1 & 2 illustrate two different but

related problems. 1. Find the intervals of concavity and

inflection points. 2. Find the local max. /min. values

Expectations Follow the steps to solve the two

problems