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Concavity & Inflection Points. Mr. Miehl miehlm@tesd.net. Objectives. To determine the intervals on which the graph of a function is concave up or concave down. To find the inflection points of a graph of a function. Concavity. - PowerPoint PPT Presentation
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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.
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