Concavity &Concavity &Inflection PointsInflection Points

Mr. MiehlMr. Miehl

miehlm@tesd.netmiehlm@tesd.net

ObjectivesObjectives

To determine the intervals on which To determine the intervals on which the graph of a function is concave up the graph of a function is concave up or concave down.or concave down.

To find the inflection points of a To find the inflection points of a graph of a function.graph of a function.

ConcavityConcavity

The The concavityconcavity of the graph of a of the graph of a function is the notion of curving function is the notion of curving upwardupward or or downwarddownward..

ConcavityConcavity

curved upwardor

concave up

ConcavityConcavity

curved downwardor

concave down

ConcavityConcavity

curved upwardor

concave up

ConcavityConcavity

Question:Question: Is the slope of the tangent Is the slope of the tangent line increasing or decreasing?line increasing or decreasing?

ConcavityConcavity

What is the derivative doing?

ConcavityConcavity

Question:Question: Is the slope of the tangent Is the slope of the tangent line increasing or decreasing?line increasing or decreasing?

Answer:Answer: The slope is increasing. The slope is increasing.

The derivative must be increasing.The derivative must be increasing.

ConcavityConcavity

Question:Question: How do we determine How do we determine where the where the derivativederivative is increasing? is increasing?

ConcavityConcavity

Question:Question: How do we determine How do we determine where a where a functionfunction is increasing? is increasing?

f f ((xx)) is increasing if is increasing if f’f’ ( (xx) > 0) > 0..

ConcavityConcavity

Question:Question: How do we determine How do we determine where the where the derivativederivative is increasing? is increasing?

f’ f’ ((xx)) is increasing if is increasing if f”f” ( (xx) > 0) > 0..

Answer:Answer: We must find where the We must find where the second derivative is positive.second derivative is positive.

ConcavityConcavity

What is the derivative doing?

ConcavityConcavity

The The concavityconcavity of a graph can be determined by of a graph can be determined by using the using the secondsecond derivativederivative..

If the If the secondsecond derivativederivative of a function is of a function is positivepositive on a given interval, then the graph of the function on a given interval, then the graph of the function is is concave upconcave up on that interval. on that interval.

If the If the secondsecond derivativederivative of a function is of a function is negativenegative on a given interval, then the graph of the function on a given interval, then the graph of the function is is concave downconcave down on that interval. on that interval.

The Second DerivativeThe Second Derivative

If If f”f” ( (xx) > 0) > 0 , , thenthen f f ((xx)) is is concaveconcave upup..

If If f”f” ( (xx) < 0) < 0 , , thenthen f f ((xx)) is is concaveconcave downdown..

ConcavityConcavity

Here the concavity changes.

This is called an inflection point (or point of inflection).

Concave down

Concave up

"( ) 0f x

"( ) 0f x

ConcavityConcavity

Concave down

"( ) 0f x

Concave up

"( ) 0f x

Inflection point

Inflection PointsInflection Points

Inflection pointsInflection points are points where are points where the graph the graph changeschanges concavity. concavity.

The second derivative will either The second derivative will either equal equal zerozero or be or be undefinedundefined at an at an inflection point.inflection point.

ConcavityConcavity

2( ) 4 16 2f x x x

'( ) 8 16f x x

''( ) 8f x

''( ) 0f x

Find the intervals on which the function is concave up or Find the intervals on which the function is concave up or concave down and the coordinates of any inflection points:concave down and the coordinates of any inflection points:

Always Concave up

ConcavityConcavity2( ) 4 16 2f x x x

Concave up: ( , )

Concave down: Never

ConcavityConcavity Find the intervals on which the function is concave up or Find the intervals on which the function is concave up or

concave down and the coordinates of any inflection points:concave down and the coordinates of any inflection points:3 2( ) 3 9 1g x x x x

2'( ) 3 6 9g x x x

"( ) 6 6g x x

0 6( 1)x

1 0x 1x

ConcavityConcavity

1

"(0) 0g "(2) 0g

0

0x

"( )g x

2x

3 2( ) 3 9 1g x x x x "( ) 6 6g x x

Concave down: ( , 1)

Concave up: (1, )

Inflection PointInflection Point3 2( ) 3 9 1g x x x x

3 2(1) (1) 3(1) 9(1) 1g

(1) 1 3 9 1g

(1) 10g

Inflection Point: (1, 10)

ConcavityConcavity3 2( ) 3 9 1g x x x x

Concave down: ( , 1)

Concave up: (1, )

Inflection Point: (1, 10)

ConcavityConcavity

13( )h x x

231

'( )3

h x x

532

"( )9

h x x

3 5

2"( )

9h x

x

Find the intervals on which the function is concave up or Find the intervals on which the function is concave up or concave down and the coordinates of any inflection points:concave down and the coordinates of any inflection points:

ConcavityConcavity

0

"( 1) 0h "(1) 0h

UND.

1x

"( )h x

1x

13( )h x x

3 5

2"( )

9h x

x

Concave up: ( , 0) Concave down: (0, )

Inflection PointInflection Point1

3( )h x x1

3(0) (0)h

(0) 0h

Inflection Point: (0, 0)

ConcavityConcavity1

3( )h x x

Concave up: ( , 0)

Concave down: (0, )

Inflection Point: (0, 0)

ConclusionConclusion

The The secondsecond derivative can be used to determine derivative can be used to determine where the graph of a function is concave up or where the graph of a function is concave up or concave down and to find concave down and to find inflectioninflection points. points.

Knowing the Knowing the criticalcritical points, increasing and points, increasing and decreasing decreasing intervalsintervals, relative , relative extremeextreme values, the values, the concavityconcavity, and the , and the inflectioninflection points of a function points of a function enables you to sketch accurate graphs of that enables you to sketch accurate graphs of that function.function.