SlideShare a Scribd company logo
Relative Extrema
First Derivative Test - FDT
Second Derivative Test - SDT
Relative Extrema
de
c

x

1

relative
maximum

re
as
ing

x

g
sin
a
cre
in
2

relative
minimum

de
cre
as

x

3

relative
maximum

ing

x

4

relative
minimum
Critical Points
Critical points are points at which:
•Derivative equals zero (also called
stationary point).
•Derivative doesn’t exist.
First Derivative Test
Let f be a differentiable function with f '(c) = 0, then:
•If f '(x) changes from positive to negative, then f has a
relative maximum at c.
•If f '(x) changes from negative to positive, then f has a
relative minimum at c.
•If f '(x) has the same sign from left to right, then f
does not have a relative extremum at c.
Practice Time!!!
Use First Derivative Test to find critical
points and state whether they are
5
2
minimums or maximums. f ( x ) = 3x 3 −15x 3
2

−1

−1

2

1

−
5 3
2 3
3
3
f ' ( x ) = 3× x −15× x
f ' ( x ) = 5x −10x = 5x 3 ( x − 2 )
3
3
1
−
5 ( x − 2)
5x 3 ( x − 2 ) = 0
Critical points
=0
1
x3

+

0

__

•
0

relative
maximum

•
2
relative
minimum

+

2(stationary)
Second Derivative Test
Suppose that c is a critical point at which f’(c) = 0,
that f(x) exists in a neighborhood of c, and that f(c)
exists. Then:
• f has a relative maximum value at c if f”(c) < 0.
•f has a relative minimum value at c if f”(c) > 0.
•If f(c) = 0, the test is not conclusive.
Note: Second derivative test is still
used to calculate max and min
Practice Time again !!!
Use second derivative test to find extrema
of f (x) = 3x 5 − 5x 3

f '(x) = 15x − 15x
4

2

15x − 15x = 0
4

2

critical points = 0, -1, 1

f "(x) = 60x 3 − 30x
inconclusive
f "(0) = 0
f "(− 1) = − 30 < 0 f has a maximum at x = -1
f has a minimum at x = 1
f "(1) = 30 > 0

15x 2 ( x 2 − 1) = 0
x2
Find extrema of f (x) =
x 4 +16
A.Using first derivative test
B.Using second derivative test

More Related Content

What's hot

Chapter 4 part3- Means and Variances of Random Variables
Chapter 4 part3- Means and Variances of Random VariablesChapter 4 part3- Means and Variances of Random Variables
Chapter 4 part3- Means and Variances of Random Variables
nszakir
 
Lesson 3.3 First Derivative Information
Lesson 3.3 First Derivative InformationLesson 3.3 First Derivative Information
Lesson 3.3 First Derivative Information
Sharon Henry
 
Lesson 19: Maximum and Minimum Values
Lesson 19: Maximum and Minimum ValuesLesson 19: Maximum and Minimum Values
Lesson 19: Maximum and Minimum Values
Matthew Leingang
 
Discrete probability distributions
Discrete probability distributionsDiscrete probability distributions
Discrete probability distributions
Nadeem Uddin
 
3.1 derivative of a function
3.1 derivative of a function3.1 derivative of a function
3.1 derivative of a function
btmathematics
 
Section 5.4 logarithmic functions
Section 5.4 logarithmic functions Section 5.4 logarithmic functions
Section 5.4 logarithmic functions
Wong Hsiung
 
Exponential and logarithmic functions
Exponential and logarithmic functionsExponential and logarithmic functions
Exponential and logarithmic functions
Njabulo Nkabinde
 
Chap05 continuous random variables and probability distributions
Chap05 continuous random variables and probability distributionsChap05 continuous random variables and probability distributions
Chap05 continuous random variables and probability distributions
Judianto Nugroho
 
Hypergeometric probability distribution
Hypergeometric probability distributionHypergeometric probability distribution
Hypergeometric probability distribution
Nadeem Uddin
 
Functions limits and continuity
Functions limits and continuityFunctions limits and continuity
Functions limits and continuity
sudersana viswanathan
 
Differentiation
DifferentiationDifferentiation
Differentiation
timschmitz
 
limits and continuity
limits and continuity limits and continuity
limits and continuity
imran khan
 
Lesson 11: Limits and Continuity
Lesson 11: Limits and ContinuityLesson 11: Limits and Continuity
Lesson 11: Limits and Continuity
Matthew Leingang
 
Obj. 41 Geometric Probability
Obj. 41 Geometric ProbabilityObj. 41 Geometric Probability
Obj. 41 Geometric Probability
smiller5
 
3.1 Extreme Values of Functions
3.1 Extreme Values of Functions3.1 Extreme Values of Functions
3.1 Extreme Values of Functions
Sharon Henry
 
Composite functions
Composite functionsComposite functions
Composite functions
Jessica Garcia
 
Factor theorem
Factor theoremFactor theorem
Factor theorem
Department of Education
 
Deriving the composition of functions
Deriving the composition of functionsDeriving the composition of functions
Deriving the composition of functions
Alona Hall
 
FM calculus
FM calculusFM calculus
FM calculus
Mike Hoad
 
Polynomial Function and Synthetic Division
Polynomial Function and Synthetic DivisionPolynomial Function and Synthetic Division
Polynomial Function and Synthetic Division
AleczQ1414
 

What's hot (20)

Chapter 4 part3- Means and Variances of Random Variables
Chapter 4 part3- Means and Variances of Random VariablesChapter 4 part3- Means and Variances of Random Variables
Chapter 4 part3- Means and Variances of Random Variables
 
Lesson 3.3 First Derivative Information
Lesson 3.3 First Derivative InformationLesson 3.3 First Derivative Information
Lesson 3.3 First Derivative Information
 
Lesson 19: Maximum and Minimum Values
Lesson 19: Maximum and Minimum ValuesLesson 19: Maximum and Minimum Values
Lesson 19: Maximum and Minimum Values
 
Discrete probability distributions
Discrete probability distributionsDiscrete probability distributions
Discrete probability distributions
 
3.1 derivative of a function
3.1 derivative of a function3.1 derivative of a function
3.1 derivative of a function
 
Section 5.4 logarithmic functions
Section 5.4 logarithmic functions Section 5.4 logarithmic functions
Section 5.4 logarithmic functions
 
Exponential and logarithmic functions
Exponential and logarithmic functionsExponential and logarithmic functions
Exponential and logarithmic functions
 
Chap05 continuous random variables and probability distributions
Chap05 continuous random variables and probability distributionsChap05 continuous random variables and probability distributions
Chap05 continuous random variables and probability distributions
 
Hypergeometric probability distribution
Hypergeometric probability distributionHypergeometric probability distribution
Hypergeometric probability distribution
 
Functions limits and continuity
Functions limits and continuityFunctions limits and continuity
Functions limits and continuity
 
Differentiation
DifferentiationDifferentiation
Differentiation
 
limits and continuity
limits and continuity limits and continuity
limits and continuity
 
Lesson 11: Limits and Continuity
Lesson 11: Limits and ContinuityLesson 11: Limits and Continuity
Lesson 11: Limits and Continuity
 
Obj. 41 Geometric Probability
Obj. 41 Geometric ProbabilityObj. 41 Geometric Probability
Obj. 41 Geometric Probability
 
3.1 Extreme Values of Functions
3.1 Extreme Values of Functions3.1 Extreme Values of Functions
3.1 Extreme Values of Functions
 
Composite functions
Composite functionsComposite functions
Composite functions
 
Factor theorem
Factor theoremFactor theorem
Factor theorem
 
Deriving the composition of functions
Deriving the composition of functionsDeriving the composition of functions
Deriving the composition of functions
 
FM calculus
FM calculusFM calculus
FM calculus
 
Polynomial Function and Synthetic Division
Polynomial Function and Synthetic DivisionPolynomial Function and Synthetic Division
Polynomial Function and Synthetic Division
 

Viewers also liked

L4 one sided limits limits at infinity
L4 one sided limits limits at infinityL4 one sided limits limits at infinity
L4 one sided limits limits at infinity
James Tagara
 
Calc 1.4a
Calc 1.4aCalc 1.4a
Calc 1.4a
hartcher
 
Chapter 02 differentiation
Chapter 02 differentiationChapter 02 differentiation
Chapter 02 differentiation
Chaïma Chaouachi
 
11 x1 t10 01 first derivative (2012)
11 x1 t10 01 first derivative (2012)11 x1 t10 01 first derivative (2012)
11 x1 t10 01 first derivative (2012)
Nigel Simmons
 
Sulpcegu5e ppt 14_3
Sulpcegu5e ppt 14_3Sulpcegu5e ppt 14_3
Sulpcegu5e ppt 14_3
silvia
 
11 x1 t12 01 first derivative (2013)
11 x1 t12 01 first derivative (2013)11 x1 t12 01 first derivative (2013)
11 x1 t12 01 first derivative (2013)
Nigel Simmons
 
Kalkulus II (5 - 6)
Kalkulus II  (5 - 6)Kalkulus II  (5 - 6)
Kalkulus II (5 - 6)
jayamartha
 
Lar calc10 ch01_sec4
Lar calc10 ch01_sec4Lar calc10 ch01_sec4
Lar calc10 ch01_sec4
Institute of Applied Technology
 
neoplastic disruptions alterations in cell function & differentiation pp
neoplastic disruptions   alterations in cell function & differentiation ppneoplastic disruptions   alterations in cell function & differentiation pp
neoplastic disruptions alterations in cell function & differentiation pp
twiggypiggy
 
Vocabulary
VocabularyVocabulary
Vocabulary
calcproject
 
IB Maths. Graphs of a funtion and its derivative
IB Maths. Graphs of a funtion and its derivativeIB Maths. Graphs of a funtion and its derivative
IB Maths. Graphs of a funtion and its derivative
estelav
 
Lesson 3: The Limit of a Function (slides)
Lesson 3: The Limit of a Function (slides)Lesson 3: The Limit of a Function (slides)
Lesson 3: The Limit of a Function (slides)
Matthew Leingang
 
Energy minimization
Energy minimizationEnergy minimization
Energy minimization
Pinky Vincent
 
Indeterminate Forms and L' Hospital Rule
Indeterminate Forms and L' Hospital RuleIndeterminate Forms and L' Hospital Rule
Indeterminate Forms and L' Hospital Rule
Aakash Singh
 
Lesson 3: The Limit of a Function
Lesson 3: The Limit of a FunctionLesson 3: The Limit of a Function
Lesson 3: The Limit of a Function
Matthew Leingang
 
Differentiation For High Ability Learners
Differentiation For High Ability LearnersDifferentiation For High Ability Learners
Differentiation For High Ability Learners
acoleman
 
Limits And Derivative
Limits And DerivativeLimits And Derivative
Limits And Derivative
Ashams kurian
 
Math major 14 differential calculus pw
Math major 14 differential calculus pwMath major 14 differential calculus pw
Math major 14 differential calculus pw
Reymart Bargamento
 
Application of differentiation
Application of differentiationApplication of differentiation
Application of differentiation
Lily Maryati
 
The Application of Derivatives
The Application of DerivativesThe Application of Derivatives
The Application of Derivatives
divaprincess09
 

Viewers also liked (20)

L4 one sided limits limits at infinity
L4 one sided limits limits at infinityL4 one sided limits limits at infinity
L4 one sided limits limits at infinity
 
Calc 1.4a
Calc 1.4aCalc 1.4a
Calc 1.4a
 
Chapter 02 differentiation
Chapter 02 differentiationChapter 02 differentiation
Chapter 02 differentiation
 
11 x1 t10 01 first derivative (2012)
11 x1 t10 01 first derivative (2012)11 x1 t10 01 first derivative (2012)
11 x1 t10 01 first derivative (2012)
 
Sulpcegu5e ppt 14_3
Sulpcegu5e ppt 14_3Sulpcegu5e ppt 14_3
Sulpcegu5e ppt 14_3
 
11 x1 t12 01 first derivative (2013)
11 x1 t12 01 first derivative (2013)11 x1 t12 01 first derivative (2013)
11 x1 t12 01 first derivative (2013)
 
Kalkulus II (5 - 6)
Kalkulus II  (5 - 6)Kalkulus II  (5 - 6)
Kalkulus II (5 - 6)
 
Lar calc10 ch01_sec4
Lar calc10 ch01_sec4Lar calc10 ch01_sec4
Lar calc10 ch01_sec4
 
neoplastic disruptions alterations in cell function & differentiation pp
neoplastic disruptions   alterations in cell function & differentiation ppneoplastic disruptions   alterations in cell function & differentiation pp
neoplastic disruptions alterations in cell function & differentiation pp
 
Vocabulary
VocabularyVocabulary
Vocabulary
 
IB Maths. Graphs of a funtion and its derivative
IB Maths. Graphs of a funtion and its derivativeIB Maths. Graphs of a funtion and its derivative
IB Maths. Graphs of a funtion and its derivative
 
Lesson 3: The Limit of a Function (slides)
Lesson 3: The Limit of a Function (slides)Lesson 3: The Limit of a Function (slides)
Lesson 3: The Limit of a Function (slides)
 
Energy minimization
Energy minimizationEnergy minimization
Energy minimization
 
Indeterminate Forms and L' Hospital Rule
Indeterminate Forms and L' Hospital RuleIndeterminate Forms and L' Hospital Rule
Indeterminate Forms and L' Hospital Rule
 
Lesson 3: The Limit of a Function
Lesson 3: The Limit of a FunctionLesson 3: The Limit of a Function
Lesson 3: The Limit of a Function
 
Differentiation For High Ability Learners
Differentiation For High Ability LearnersDifferentiation For High Ability Learners
Differentiation For High Ability Learners
 
Limits And Derivative
Limits And DerivativeLimits And Derivative
Limits And Derivative
 
Math major 14 differential calculus pw
Math major 14 differential calculus pwMath major 14 differential calculus pw
Math major 14 differential calculus pw
 
Application of differentiation
Application of differentiationApplication of differentiation
Application of differentiation
 
The Application of Derivatives
The Application of DerivativesThe Application of Derivatives
The Application of Derivatives
 

Similar to 5.2 first and second derivative test

Calc 3.4b
Calc 3.4bCalc 3.4b
Calc 3.4b
hartcher
 
Day 1a examples
Day 1a examplesDay 1a examples
Day 1a examples
jchartiersjsd
 
Calc 3.4
Calc 3.4Calc 3.4
Calc 3.4
hartcher
 
Derivatives in graphing-dfs
Derivatives in graphing-dfsDerivatives in graphing-dfs
Derivatives in graphing-dfs
Farhana Shaheen
 
Curve sketching
Curve sketchingCurve sketching
Curve sketching
Rnold Wilson
 
Ap calculus extrema v2
Ap calculus extrema v2Ap calculus extrema v2
Ap calculus extrema v2
gregcross22
 
Mac2311 study guide-tcm6-49721
Mac2311 study guide-tcm6-49721Mac2311 study guide-tcm6-49721
Mac2311 study guide-tcm6-49721
Glicerio Gavilan
 
2301MaxMin.ppt
2301MaxMin.ppt2301MaxMin.ppt
2301MaxMin.ppt
AsadAltaf10
 
Probability-1.pptx
Probability-1.pptxProbability-1.pptx
Probability-1.pptx
MuskanKhan320706
 
Applications of Differentiation
Applications of DifferentiationApplications of Differentiation
Applications of Differentiation
Joey Valdriz
 
Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)
Mel Anthony Pepito
 
Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)
Matthew Leingang
 
Limit and continuity
Limit and continuityLimit and continuity
Limit and continuity
Digvijaysinh Gohil
 
Maximums and minimum
Maximums and minimum Maximums and minimum
Maximums and minimum
rubimedina01
 
Lecture 12(point of inflection and concavity)
Lecture 12(point of inflection and concavity)Lecture 12(point of inflection and concavity)
Lecture 12(point of inflection and concavity)
FahadYaqoob5
 
Lecture 12(point of inflection and concavity)
Lecture 12(point of inflection and concavity)Lecture 12(point of inflection and concavity)
Lecture 12(point of inflection and concavity)
FahadYaqoob5
 
Calculus
CalculusCalculus
Calculus
Jay Bhatt
 
Application of derivative
Application of derivativeApplication of derivative
Application of derivatives
Application of derivativesApplication of derivatives
Application of derivatives
indu thakur
 
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
Lesson 3.3 3.4 - 1 st and 2nd Derivative InformationLesson 3.3 3.4 - 1 st and 2nd Derivative Information
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
Sharon Henry
 

Similar to 5.2 first and second derivative test (20)

Calc 3.4b
Calc 3.4bCalc 3.4b
Calc 3.4b
 
Day 1a examples
Day 1a examplesDay 1a examples
Day 1a examples
 
Calc 3.4
Calc 3.4Calc 3.4
Calc 3.4
 
Derivatives in graphing-dfs
Derivatives in graphing-dfsDerivatives in graphing-dfs
Derivatives in graphing-dfs
 
Curve sketching
Curve sketchingCurve sketching
Curve sketching
 
Ap calculus extrema v2
Ap calculus extrema v2Ap calculus extrema v2
Ap calculus extrema v2
 
Mac2311 study guide-tcm6-49721
Mac2311 study guide-tcm6-49721Mac2311 study guide-tcm6-49721
Mac2311 study guide-tcm6-49721
 
2301MaxMin.ppt
2301MaxMin.ppt2301MaxMin.ppt
2301MaxMin.ppt
 
Probability-1.pptx
Probability-1.pptxProbability-1.pptx
Probability-1.pptx
 
Applications of Differentiation
Applications of DifferentiationApplications of Differentiation
Applications of Differentiation
 
Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)
 
Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)
 
Limit and continuity
Limit and continuityLimit and continuity
Limit and continuity
 
Maximums and minimum
Maximums and minimum Maximums and minimum
Maximums and minimum
 
Lecture 12(point of inflection and concavity)
Lecture 12(point of inflection and concavity)Lecture 12(point of inflection and concavity)
Lecture 12(point of inflection and concavity)
 
Lecture 12(point of inflection and concavity)
Lecture 12(point of inflection and concavity)Lecture 12(point of inflection and concavity)
Lecture 12(point of inflection and concavity)
 
Calculus
CalculusCalculus
Calculus
 
Application of derivative
Application of derivativeApplication of derivative
Application of derivative
 
Application of derivatives
Application of derivativesApplication of derivatives
Application of derivatives
 
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
Lesson 3.3 3.4 - 1 st and 2nd Derivative InformationLesson 3.3 3.4 - 1 st and 2nd Derivative Information
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
 

More from dicosmo178

8.7 numerical integration
8.7 numerical integration8.7 numerical integration
8.7 numerical integration
dicosmo178
 
8.2 integration by parts
8.2 integration by parts8.2 integration by parts
8.2 integration by parts
dicosmo178
 
7.3 volumes by cylindrical shells
7.3 volumes by cylindrical shells7.3 volumes by cylindrical shells
7.3 volumes by cylindrical shells
dicosmo178
 
7.2 volumes by slicing disks and washers
7.2 volumes by slicing disks and washers7.2 volumes by slicing disks and washers
7.2 volumes by slicing disks and washers
dicosmo178
 
7.1 area between curves
7.1 area between curves7.1 area between curves
7.1 area between curves
dicosmo178
 
6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus...
6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus...6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus...
6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus...dicosmo178
 
6.3 integration by substitution
6.3 integration by substitution6.3 integration by substitution
6.3 integration by substitution
dicosmo178
 
6.2 the indefinite integral
6.2 the indefinite integral 6.2 the indefinite integral
6.2 the indefinite integral
dicosmo178
 
6.1 & 6.4 an overview of the area problem area
6.1 & 6.4 an overview of the area problem area6.1 & 6.4 an overview of the area problem area
6.1 & 6.4 an overview of the area problem area
dicosmo178
 
5.8 rectilinear motion
5.8 rectilinear motion5.8 rectilinear motion
5.8 rectilinear motion
dicosmo178
 
5.7 rolle's thrm & mv theorem
5.7 rolle's thrm & mv theorem5.7 rolle's thrm & mv theorem
5.7 rolle's thrm & mv theorem
dicosmo178
 
5.5 optimization
5.5 optimization5.5 optimization
5.5 optimization
dicosmo178
 
5.4 absolute maxima and minima
5.4 absolute maxima and minima5.4 absolute maxima and minima
5.4 absolute maxima and minima
dicosmo178
 
5.3 curve sketching
5.3 curve sketching5.3 curve sketching
5.3 curve sketching
dicosmo178
 
5.1 analysis of function i
5.1 analysis of function i5.1 analysis of function i
5.1 analysis of function i
dicosmo178
 
4.3 derivatives of inv erse trig. functions
4.3 derivatives of inv erse trig. functions4.3 derivatives of inv erse trig. functions
4.3 derivatives of inv erse trig. functions
dicosmo178
 
7.2 volumes by slicing disks and washers
7.2 volumes by slicing disks and washers7.2 volumes by slicing disks and washers
7.2 volumes by slicing disks and washersdicosmo178
 
8.2 integration by parts
8.2 integration by parts8.2 integration by parts
8.2 integration by parts
dicosmo178
 
8.7 numerical integration
8.7 numerical integration8.7 numerical integration
8.7 numerical integration
dicosmo178
 
7.3 volumes by cylindrical shells
7.3 volumes by cylindrical shells7.3 volumes by cylindrical shells
7.3 volumes by cylindrical shells
dicosmo178
 

More from dicosmo178 (20)

8.7 numerical integration
8.7 numerical integration8.7 numerical integration
8.7 numerical integration
 
8.2 integration by parts
8.2 integration by parts8.2 integration by parts
8.2 integration by parts
 
7.3 volumes by cylindrical shells
7.3 volumes by cylindrical shells7.3 volumes by cylindrical shells
7.3 volumes by cylindrical shells
 
7.2 volumes by slicing disks and washers
7.2 volumes by slicing disks and washers7.2 volumes by slicing disks and washers
7.2 volumes by slicing disks and washers
 
7.1 area between curves
7.1 area between curves7.1 area between curves
7.1 area between curves
 
6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus...
6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus...6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus...
6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus...
 
6.3 integration by substitution
6.3 integration by substitution6.3 integration by substitution
6.3 integration by substitution
 
6.2 the indefinite integral
6.2 the indefinite integral 6.2 the indefinite integral
6.2 the indefinite integral
 
6.1 & 6.4 an overview of the area problem area
6.1 & 6.4 an overview of the area problem area6.1 & 6.4 an overview of the area problem area
6.1 & 6.4 an overview of the area problem area
 
5.8 rectilinear motion
5.8 rectilinear motion5.8 rectilinear motion
5.8 rectilinear motion
 
5.7 rolle's thrm & mv theorem
5.7 rolle's thrm & mv theorem5.7 rolle's thrm & mv theorem
5.7 rolle's thrm & mv theorem
 
5.5 optimization
5.5 optimization5.5 optimization
5.5 optimization
 
5.4 absolute maxima and minima
5.4 absolute maxima and minima5.4 absolute maxima and minima
5.4 absolute maxima and minima
 
5.3 curve sketching
5.3 curve sketching5.3 curve sketching
5.3 curve sketching
 
5.1 analysis of function i
5.1 analysis of function i5.1 analysis of function i
5.1 analysis of function i
 
4.3 derivatives of inv erse trig. functions
4.3 derivatives of inv erse trig. functions4.3 derivatives of inv erse trig. functions
4.3 derivatives of inv erse trig. functions
 
7.2 volumes by slicing disks and washers
7.2 volumes by slicing disks and washers7.2 volumes by slicing disks and washers
7.2 volumes by slicing disks and washers
 
8.2 integration by parts
8.2 integration by parts8.2 integration by parts
8.2 integration by parts
 
8.7 numerical integration
8.7 numerical integration8.7 numerical integration
8.7 numerical integration
 
7.3 volumes by cylindrical shells
7.3 volumes by cylindrical shells7.3 volumes by cylindrical shells
7.3 volumes by cylindrical shells
 

Recently uploaded

Authentically Social Presented by Corey Perlman
Authentically Social Presented by Corey PerlmanAuthentically Social Presented by Corey Perlman
Authentically Social Presented by Corey Perlman
Corey Perlman, Social Media Speaker and Consultant
 
ikea_woodgreen_petscharity_cat-alogue_digital.pdf
ikea_woodgreen_petscharity_cat-alogue_digital.pdfikea_woodgreen_petscharity_cat-alogue_digital.pdf
ikea_woodgreen_petscharity_cat-alogue_digital.pdf
agatadrynko
 
Call 8867766396 Satta Matka Dpboss Matka Guessing Satta batta Matka 420 Satta...
Call 8867766396 Satta Matka Dpboss Matka Guessing Satta batta Matka 420 Satta...Call 8867766396 Satta Matka Dpboss Matka Guessing Satta batta Matka 420 Satta...
Call 8867766396 Satta Matka Dpboss Matka Guessing Satta batta Matka 420 Satta...
bosssp10
 
一比一原版新西兰奥塔哥大学毕业证(otago毕业证)如何办理
一比一原版新西兰奥塔哥大学毕业证(otago毕业证)如何办理一比一原版新西兰奥塔哥大学毕业证(otago毕业证)如何办理
一比一原版新西兰奥塔哥大学毕业证(otago毕业证)如何办理
taqyea
 
Satta Matka Dpboss Matka Guessing Kalyan Chart Indian Matka Kalyan panel Chart
Satta Matka Dpboss Matka Guessing Kalyan Chart Indian Matka Kalyan panel ChartSatta Matka Dpboss Matka Guessing Kalyan Chart Indian Matka Kalyan panel Chart
Satta Matka Dpboss Matka Guessing Kalyan Chart Indian Matka Kalyan panel Chart
➒➌➎➏➑➐➋➑➐➐Dpboss Matka Guessing Satta Matka Kalyan Chart Indian Matka
 
Industrial Tech SW: Category Renewal and Creation
Industrial Tech SW:  Category Renewal and CreationIndustrial Tech SW:  Category Renewal and Creation
Industrial Tech SW: Category Renewal and Creation
Christian Dahlen
 
3 Simple Steps To Buy Verified Payoneer Account In 2024
3 Simple Steps To Buy Verified Payoneer Account In 20243 Simple Steps To Buy Verified Payoneer Account In 2024
3 Simple Steps To Buy Verified Payoneer Account In 2024
SEOSMMEARTH
 
Authentically Social by Corey Perlman - EO Puerto Rico
Authentically Social by Corey Perlman - EO Puerto RicoAuthentically Social by Corey Perlman - EO Puerto Rico
Authentically Social by Corey Perlman - EO Puerto Rico
Corey Perlman, Social Media Speaker and Consultant
 
Mastering B2B Payments Webinar from BlueSnap
Mastering B2B Payments Webinar from BlueSnapMastering B2B Payments Webinar from BlueSnap
Mastering B2B Payments Webinar from BlueSnap
Norma Mushkat Gaffin
 
Organizational Change Leadership Agile Tour Geneve 2024
Organizational Change Leadership Agile Tour Geneve 2024Organizational Change Leadership Agile Tour Geneve 2024
Organizational Change Leadership Agile Tour Geneve 2024
Kirill Klimov
 
Building Your Employer Brand with Social Media
Building Your Employer Brand with Social MediaBuilding Your Employer Brand with Social Media
Building Your Employer Brand with Social Media
LuanWise
 
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s DholeraTata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera
Avirahi City Dholera
 
Unveiling the Dynamic Personalities, Key Dates, and Horoscope Insights: Gemin...
Unveiling the Dynamic Personalities, Key Dates, and Horoscope Insights: Gemin...Unveiling the Dynamic Personalities, Key Dates, and Horoscope Insights: Gemin...
Unveiling the Dynamic Personalities, Key Dates, and Horoscope Insights: Gemin...
my Pandit
 
Dpboss Matka Guessing Satta Matta Matka Kalyan Chart Satta Matka
Dpboss Matka Guessing Satta Matta Matka Kalyan Chart Satta MatkaDpboss Matka Guessing Satta Matta Matka Kalyan Chart Satta Matka
Dpboss Matka Guessing Satta Matta Matka Kalyan Chart Satta Matka
➒➌➎➏➑➐➋➑➐➐Dpboss Matka Guessing Satta Matka Kalyan Chart Indian Matka
 
Creative Web Design Company in Singapore
Creative Web Design Company in SingaporeCreative Web Design Company in Singapore
Creative Web Design Company in Singapore
techboxsqauremedia
 
amptalk_RecruitingDeck_english_2024.06.05
amptalk_RecruitingDeck_english_2024.06.05amptalk_RecruitingDeck_english_2024.06.05
amptalk_RecruitingDeck_english_2024.06.05
marketing317746
 
Recruiting in the Digital Age: A Social Media Masterclass
Recruiting in the Digital Age: A Social Media MasterclassRecruiting in the Digital Age: A Social Media Masterclass
Recruiting in the Digital Age: A Social Media Masterclass
LuanWise
 
The Influence of Marketing Strategy and Market Competition on Business Perfor...
The Influence of Marketing Strategy and Market Competition on Business Perfor...The Influence of Marketing Strategy and Market Competition on Business Perfor...
The Influence of Marketing Strategy and Market Competition on Business Perfor...
Adam Smith
 
The Evolution and Impact of OTT Platforms: A Deep Dive into the Future of Ent...
The Evolution and Impact of OTT Platforms: A Deep Dive into the Future of Ent...The Evolution and Impact of OTT Platforms: A Deep Dive into the Future of Ent...
The Evolution and Impact of OTT Platforms: A Deep Dive into the Future of Ent...
ABHILASH DUTTA
 
Company Valuation webinar series - Tuesday, 4 June 2024
Company Valuation webinar series - Tuesday, 4 June 2024Company Valuation webinar series - Tuesday, 4 June 2024
Company Valuation webinar series - Tuesday, 4 June 2024
FelixPerez547899
 

Recently uploaded (20)

Authentically Social Presented by Corey Perlman
Authentically Social Presented by Corey PerlmanAuthentically Social Presented by Corey Perlman
Authentically Social Presented by Corey Perlman
 
ikea_woodgreen_petscharity_cat-alogue_digital.pdf
ikea_woodgreen_petscharity_cat-alogue_digital.pdfikea_woodgreen_petscharity_cat-alogue_digital.pdf
ikea_woodgreen_petscharity_cat-alogue_digital.pdf
 
Call 8867766396 Satta Matka Dpboss Matka Guessing Satta batta Matka 420 Satta...
Call 8867766396 Satta Matka Dpboss Matka Guessing Satta batta Matka 420 Satta...Call 8867766396 Satta Matka Dpboss Matka Guessing Satta batta Matka 420 Satta...
Call 8867766396 Satta Matka Dpboss Matka Guessing Satta batta Matka 420 Satta...
 
一比一原版新西兰奥塔哥大学毕业证(otago毕业证)如何办理
一比一原版新西兰奥塔哥大学毕业证(otago毕业证)如何办理一比一原版新西兰奥塔哥大学毕业证(otago毕业证)如何办理
一比一原版新西兰奥塔哥大学毕业证(otago毕业证)如何办理
 
Satta Matka Dpboss Matka Guessing Kalyan Chart Indian Matka Kalyan panel Chart
Satta Matka Dpboss Matka Guessing Kalyan Chart Indian Matka Kalyan panel ChartSatta Matka Dpboss Matka Guessing Kalyan Chart Indian Matka Kalyan panel Chart
Satta Matka Dpboss Matka Guessing Kalyan Chart Indian Matka Kalyan panel Chart
 
Industrial Tech SW: Category Renewal and Creation
Industrial Tech SW:  Category Renewal and CreationIndustrial Tech SW:  Category Renewal and Creation
Industrial Tech SW: Category Renewal and Creation
 
3 Simple Steps To Buy Verified Payoneer Account In 2024
3 Simple Steps To Buy Verified Payoneer Account In 20243 Simple Steps To Buy Verified Payoneer Account In 2024
3 Simple Steps To Buy Verified Payoneer Account In 2024
 
Authentically Social by Corey Perlman - EO Puerto Rico
Authentically Social by Corey Perlman - EO Puerto RicoAuthentically Social by Corey Perlman - EO Puerto Rico
Authentically Social by Corey Perlman - EO Puerto Rico
 
Mastering B2B Payments Webinar from BlueSnap
Mastering B2B Payments Webinar from BlueSnapMastering B2B Payments Webinar from BlueSnap
Mastering B2B Payments Webinar from BlueSnap
 
Organizational Change Leadership Agile Tour Geneve 2024
Organizational Change Leadership Agile Tour Geneve 2024Organizational Change Leadership Agile Tour Geneve 2024
Organizational Change Leadership Agile Tour Geneve 2024
 
Building Your Employer Brand with Social Media
Building Your Employer Brand with Social MediaBuilding Your Employer Brand with Social Media
Building Your Employer Brand with Social Media
 
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s DholeraTata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera
 
Unveiling the Dynamic Personalities, Key Dates, and Horoscope Insights: Gemin...
Unveiling the Dynamic Personalities, Key Dates, and Horoscope Insights: Gemin...Unveiling the Dynamic Personalities, Key Dates, and Horoscope Insights: Gemin...
Unveiling the Dynamic Personalities, Key Dates, and Horoscope Insights: Gemin...
 
Dpboss Matka Guessing Satta Matta Matka Kalyan Chart Satta Matka
Dpboss Matka Guessing Satta Matta Matka Kalyan Chart Satta MatkaDpboss Matka Guessing Satta Matta Matka Kalyan Chart Satta Matka
Dpboss Matka Guessing Satta Matta Matka Kalyan Chart Satta Matka
 
Creative Web Design Company in Singapore
Creative Web Design Company in SingaporeCreative Web Design Company in Singapore
Creative Web Design Company in Singapore
 
amptalk_RecruitingDeck_english_2024.06.05
amptalk_RecruitingDeck_english_2024.06.05amptalk_RecruitingDeck_english_2024.06.05
amptalk_RecruitingDeck_english_2024.06.05
 
Recruiting in the Digital Age: A Social Media Masterclass
Recruiting in the Digital Age: A Social Media MasterclassRecruiting in the Digital Age: A Social Media Masterclass
Recruiting in the Digital Age: A Social Media Masterclass
 
The Influence of Marketing Strategy and Market Competition on Business Perfor...
The Influence of Marketing Strategy and Market Competition on Business Perfor...The Influence of Marketing Strategy and Market Competition on Business Perfor...
The Influence of Marketing Strategy and Market Competition on Business Perfor...
 
The Evolution and Impact of OTT Platforms: A Deep Dive into the Future of Ent...
The Evolution and Impact of OTT Platforms: A Deep Dive into the Future of Ent...The Evolution and Impact of OTT Platforms: A Deep Dive into the Future of Ent...
The Evolution and Impact of OTT Platforms: A Deep Dive into the Future of Ent...
 
Company Valuation webinar series - Tuesday, 4 June 2024
Company Valuation webinar series - Tuesday, 4 June 2024Company Valuation webinar series - Tuesday, 4 June 2024
Company Valuation webinar series - Tuesday, 4 June 2024
 

5.2 first and second derivative test

  • 1. Relative Extrema First Derivative Test - FDT Second Derivative Test - SDT
  • 3. Critical Points Critical points are points at which: •Derivative equals zero (also called stationary point). •Derivative doesn’t exist.
  • 4. First Derivative Test Let f be a differentiable function with f '(c) = 0, then: •If f '(x) changes from positive to negative, then f has a relative maximum at c. •If f '(x) changes from negative to positive, then f has a relative minimum at c. •If f '(x) has the same sign from left to right, then f does not have a relative extremum at c.
  • 5. Practice Time!!! Use First Derivative Test to find critical points and state whether they are 5 2 minimums or maximums. f ( x ) = 3x 3 −15x 3 2 −1 −1 2 1 − 5 3 2 3 3 3 f ' ( x ) = 3× x −15× x f ' ( x ) = 5x −10x = 5x 3 ( x − 2 ) 3 3 1 − 5 ( x − 2) 5x 3 ( x − 2 ) = 0 Critical points =0 1 x3 + 0 __ • 0 relative maximum • 2 relative minimum + 2(stationary)
  • 6. Second Derivative Test Suppose that c is a critical point at which f’(c) = 0, that f(x) exists in a neighborhood of c, and that f(c) exists. Then: • f has a relative maximum value at c if f”(c) < 0. •f has a relative minimum value at c if f”(c) > 0. •If f(c) = 0, the test is not conclusive. Note: Second derivative test is still used to calculate max and min
  • 7. Practice Time again !!! Use second derivative test to find extrema of f (x) = 3x 5 − 5x 3 f '(x) = 15x − 15x 4 2 15x − 15x = 0 4 2 critical points = 0, -1, 1 f "(x) = 60x 3 − 30x inconclusive f "(0) = 0 f "(− 1) = − 30 < 0 f has a maximum at x = -1 f has a minimum at x = 1 f "(1) = 30 > 0 15x 2 ( x 2 − 1) = 0
  • 8. x2 Find extrema of f (x) = x 4 +16 A.Using first derivative test B.Using second derivative test