SLecture4 Maxima Minima

MA 105 Calculus: Lecture 4Maxima/Minima

S. Sivaji GaneshMathematics Department

IIT Bombay

July 30, 2009

S. Sivaji Ganesh (IIT Bombay) MA 105 Calculus: Lecture 4 July 30, 2009 1 / 21

Maxima/Minima: Why are they important?



1 Why a water drop is always spherical?



1 Why a water drop is always spherical?2 Why a businessman, an investor do, whatever they do?



1 Why a water drop is always spherical?2 Why a businessman, an investor do, whatever they do?3 Why Snow flakes are flat?



1 Why a water drop is always spherical?2 Why a businessman, an investor do, whatever they do?3 Why Snow flakes are flat?4 Why Aero-dynamic buses?



1 Why a water drop is always spherical?2 Why a businessman, an investor do, whatever they do?3 Why Snow flakes are flat?4 Why Aero-dynamic buses?5 In Engineering design problems, Optimization plays an important

role.




role.6 Didos problem: Given a closed rope, which is to be used for

marking boundary so that enclosed area is the largest possible





marking boundary so that enclosed area is the largest possible7 Brachistochrone: (find) The shape of the curve down which a

bead will slip from rest, accelerates and moves under the gravitywitout friction, from one point to another in the shortest time.

Mathematical analysis of maxima/minima goes under the name ofcalculus of variations.





marking boundary so that enclosed area is the largest possible7 Brachistochrone: (find) The shape of the curve down which a

bead will slip from rest, accelerates and moves under the gravitywitout friction, from one point to another in the shortest time.

Mathematical analysis of maxima/minima goes under the name ofcalculus of variations. For stories on maxima/minima, read the book byV.M. Tikhomirov on Stories About Maxima and Minima


Absolute Maximum & Absolute Minimum

Definition1 A function f has an absoute maximum (or global maximum) at c if




f (c) f (x) for all x in D, where D is the domain of f .




f (c) f (x) for all x in D, where D is the domain of f . The numberf (c) is called the maximum value




f (c) f (x) for all x in D, where D is the domain of f . The numberf (c) is called the maximum value of f




f (c) f (x) for all x in D, where D is the domain of f . The numberf (c) is called the maximum value of f on D.





2 A function f has an absoute minimum (or global minimum) at c if





2 A function f has an absoute minimum (or global minimum) at c iff (c) f (x) for all x in D.





2 A function f has an absoute minimum (or global minimum) at c iff (c) f (x) for all x in D. The number f (c) is called the minimumvalue





2 A function f has an absoute minimum (or global minimum) at c iff (c) f (x) for all x in D. The number f (c) is called the minimumvalue of f





2 A function f has an absoute minimum (or global minimum) at c iff (c) f (x) for all x in D. The number f (c) is called the minimumvalue of f on D.






3 The maximum and minimum values of f are called extreme valuesof f on D.






3 The maximum and minimum values of f are called extreme valuesof f on D.

Absolute Maximum and Absolute Minimum of a function on its Domainare the Highest and Lowest points on its graph.



What is the absolute maximum and absolute minimum for the functionwhose graph is below.



What is the absolute maximum and absolute minimum for the functionwhose graph is below. On which domain?


Local Maximum & Local Minimum

Definition1 A function f has an local maximum (or relative maximum) at c if




f (c) f (x) when x is near c




f (c) f (x) when x is near c (This meansf (c) f (x) for all x insome open interval containing c).





2 A function f has an local minimum (or relative minimum) at c if





2 A function f has an local minimum (or relative minimum) at c iff (c) f (x) when x is near c.






Note The domain of the function does not play any role in the conceptof local maximum/minimum






Note The domain of the function does not play any role in the conceptof local maximum/minimum Compare with definition of absolutemaximum/minimum definition.


The Extreme Value Theorem



Note that in all the previous examples, extreme values of the functionare taken.



Note that in all the previous examples, extreme values of the functionare taken. For continuous functions on closed intervals this can beproved.




Theorem (Extreme Value Theorem)If f is continuous on a closed interval [a,b],




Theorem (Extreme Value Theorem)If f is continuous on a closed interval [a,b], then f attains an absolutemaximum value f (c) and an absolute minimum value f (d) at somenumbers c and d in [a,b].





RemarksThe proof is difficult and we omit the proof.





RemarksThe proof is difficult and we omit the proof.An extreme value can be take on more than once.





RemarksThe proof is difficult and we omit the proof.An extreme value can be take on more than once. See sin x on[0,4pi].





RemarksThe proof is difficult and we omit the proof.An extreme value can be take on more than once. See sin x on[0,4pi]. Draw some more graphs!


The Extreme Value Theorem (contd.)



Both the hypothesis function is continuous and interval is closedcannot be omitted.



Both the hypothesis function is continuous and interval is closedcannot be omitted. If any one of them is dropped from the hypothesisof Extreme Value theorem,



Both the hypothesis function is continuous and interval is closedcannot be omitted. If any one of them is dropped from the hypothesisof Extreme Value theorem, the theorem is no longer true.




1 Revisit the previous examples,




1 Revisit the previous examples, and construct more examples toillustrate the importance of both hypotheses.





2 Converse questions?





2 Converse questions?3 Give an example of a discontinuous function on a closed interval

which verifies the conclusion of EVT.






which verifies the conclusion of EVT.4 Give an example of a continuous function on an interval which is

not closed,






which verifies the conclusion of EVT.4 Give an example of a continuous function on an interval which is

not closed, and the coclusion of EVT does not hold.


Maxima/minima


Maxima/minima

Question: Look at the examples that you constructed.


Maxima/minima

Question: Look at the examples that you constructed. Point out pointsof local maxima and local minima.


Maxima/minima

Question: Look at the examples that you constructed. Point out pointsof local maxima and local minima. What is the slope of the tangents atthose points?


Maxima/minima


Theorem (Fermats Theorem)If f has a local maximum or minimum at c,


Maxima/minima


Theorem (Fermats Theorem)If f has a local maximum or minimum at c, and if f (c) exists,


Maxima/minima


Theorem (Fermats Theorem)If f has a local maximum or minimum at c, and if f (c) exists, thenf (c) = 0.


Maxima/minima



Proof: Suppose that f has a local maximum at c.


Maxima/minima



Proof: Suppose that f has a local maximum at c. Then NewtonQuotients f (c+h)f (c)h are non-positive ( 0) if h > 0.


Maxima/minima



Proof: Suppose that f has a local maximum at c. Then NewtonQuotients f (c+h)f (c)h are non-positive ( 0) if h > 0. Thereforelimh0+ f (c+h)f (c)h 0.


Maxima/minima



Proof: Suppose that f has a local maximum at c. Then NewtonQuotients f (c+h)f (c)h are non-positive ( 0) if h > 0. Thereforelimh0+ f (c+h)f (c)h 0. Similarly, if h < 0, then

f (c+h)f (c)h are

non-negitive ( 0) if h < 0.


Maxima/minima




f (c+h)f (c)h are

non-negitive ( 0) if h < 0. Therefore limh0 f (c+h)f (c)h 0.


Maxima/minima




f (c+h)f (c)h are

non-negitive ( 0) if h < 0. Therefore limh0 f (c+h)f (c)h 0. But weare given that f is differentiable at c.


Maxima/minima




f (c+h)f (c)h are

non-negitive ( 0) if h < 0. Therefore limh0 f (c+h)f (c)h 0. But weare given that f is differentiable at c. Therefore, the left hand and righthand limits must be equal,


Maxima/minima




f (c+h)f (c)h are

non-negitive ( 0) if h < 0. Therefore limh0 f (c+h)f (c)h 0. But weare given that f is differentiable at c. Therefore, the left hand and righthand limits must be equal, and,


Maxima/minima




f (c+h)f (c)h are

non-negitive ( 0) if h < 0. Therefore limh0 f (c+h)f (c)h 0. But weare given that f is differentiable at c. Therefore, the left hand and righthand limits must be equal, and, the only way is that both limits must bezero.


Maxima/minima




f (c+h)f (c)h are

non-negitive ( 0) if h < 0. Therefore limh0 f (c+h)f (c)h 0. But weare given that f is differentiable at c. Therefore, the left hand and righthand limits must be equal, and, the only way is that both limits must bezero. This finishes the proof.


Comments on Fermats theorem



1 If f (c) = 0, then



1 If f (c) = 0, then f (c) may not be a local extremum value.



1 If f (c) = 0, then f (c) may not be a local extremum value.Therefore we cannot simply solve f (x) = 0 and say that thesolutions give local extrema.



1 If f (c) = 0, then f (c) may not be a local extremum value.Therefore we cannot simply solve f (x) = 0 and say that thesolutions give local extrema. Example?



1 If f (c) = 0, then f (c) may not be a local extremum value.Therefore we cannot simply solve f (x) = 0 and say that thesolutions give local extrema. Example? f (x) = x3




2 Note that, at a point where a local extremum value is attained,




2 Note that, at a point where a local extremum value is attained, thefunction need not be differentiable.




2 Note that, at a point where a local extremum value is attained, thefunction need not be differentiable. Give an example.





3 Summarizing,





3 Summarizing, at a point where a local extremum value is attained,





3 Summarizing, at a point where a local extremum value is attained,either function is not differentiable





3 Summarizing, at a point where a local extremum value is attained,either function is not differentiable or the derivative is zero.






DefinitionA critical number of a function f is






DefinitionA critical number of a function f is a number c in the domain of f suchthat






DefinitionA critical number of a function f is a number c in the domain of f suchthat either f (c) = 0 or f (c) does not exist.

Exercise Find the critical numbers of x3/5(4 x).






DefinitionA critical number of a function f is a number c in the domain of f suchthat either f (c) = 0 or f (c) does not exist.

Exercise Find the critical numbers of x3/5(4 x). Ans: 0 and 32 .


The Closed Interval Method to find global extrema



Let f be a continuous function on [a,b].



Let f be a continuous function on [a,b].Question: How to find the absolute maximum and minimum values off



Let f be a continuous function on [a,b].Question: How to find the absolute maximum and minimum values off on [a,b]?

1 Find the values of f at the critical numbers of f in (a,b)




1 Find the values of f at the critical numbers of f in (a,b)2 Find the values of f at the end points of the interval




1 Find the values of f at the critical numbers of f in (a,b)2 Find the values of f at the end points of the interval3 The largest (smallest) values from Steps 1 and 2 is




1 Find the values of f at the critical numbers of f in (a,b)2 Find the values of f at the end points of the interval3 The largest (smallest) values from Steps 1 and 2 is the absolute

maximum (resp. minimum) value.






Find absolute maximum and absolute minimum of f (x) = x3 3x2 + 1on [1/2,4].

0 First verify that the Closed interval method can be applied!







0 First verify that the Closed interval method can be applied!1 Find critical points.







0 First verify that the Closed interval method can be applied!1 Find critical points. They are 0 and 2.







0 First verify that the Closed interval method can be applied!1 Find critical points. They are 0 and 2. f (0) = 1 and f (2) = 3.







0 First verify that the Closed interval method can be applied!1 Find critical points. They are 0 and 2. f (0) = 1 and f (2) = 3.2 f (1/2) = 1/8 and f (4) = 17







0 First verify that the Closed interval method can be applied!1 Find critical points. They are 0 and 2. f (0) = 1 and f (2) = 3.2 f (1/2) = 1/8 and f (4) = 173 The values from Steps 1 and 2 are {1,3,1/8,17}.


Graph of f (x) = x3 3x2 + 1 on [1/2,4]


Comments on the Closed Interval Method to find global extrema



Let f be a continuous function on [a,b].



Let f be a continuous function on [a,b]. is a crucial assumption.



Let f be a continuous function on [a,b]. is a crucial assumption. Onecan not drop the continuity assumption.



Let f be a continuous function on [a,b]. is a crucial assumption. Onecan not drop the continuity assumption. Note that the steps of the Theclosed interval method can be followed for any function.



Let f be a continuous function on [a,b]. is a crucial assumption. Onecan not drop the continuity assumption. Note that the steps of the Theclosed interval method can be followed for any function. But there is noguarantee that



Let f be a continuous function on [a,b]. is a crucial assumption. Onecan not drop the continuity assumption. Note that the steps of the Theclosed interval method can be followed for any function. But there is noguarantee that whatever we conclude by following this procedure isCORRECT.Question: Find functions defined on the closed interval [0,1] that arenot continuous



Let f be a continuous function on [a,b]. is a crucial assumption. Onecan not drop the continuity assumption. Note that the steps of the Theclosed interval method can be followed for any function. But there is noguarantee that whatever we conclude by following this procedure isCORRECT.Question: Find functions defined on the closed interval [0,1] that arenot continuous (drawing its graph is enough)



Let f be a continuous function on [a,b]. is a crucial assumption. Onecan not drop the continuity assumption. Note that the steps of the Theclosed interval method can be followed for any function. But there is noguarantee that whatever we conclude by following this procedure isCORRECT.Question: Find functions defined on the closed interval [0,1] that arenot continuous (drawing its graph is enough) with the property that theconclusions we make by following the closed interval method areWRONG.




Caution: Even for continuous functions,




Caution: Even for continuous functions, the closed interval methodworks




Caution: Even for continuous functions, the closed interval methodworks because we have




Caution: Even for continuous functions, the closed interval methodworks because we have Extreme value theorem.




Caution: Even for continuous functions, the closed interval methodworks because we have Extreme value theorem. If function is notcontinuous,




Caution: Even for continuous functions, the closed interval methodworks because we have Extreme value theorem. If function is notcontinuous, then EVT cannot be invoked,




Caution: Even for continuous functions, the closed interval methodworks because we have Extreme value theorem. If function is notcontinuous, then EVT cannot be invoked, as a consequence, we donot know the existence of extreme values for the function.




Caution: Even for continuous functions, the closed interval methodworks because we have Extreme value theorem. If function is notcontinuous, then EVT cannot be invoked, as a consequence, we donot know the existence of extreme values for the function. Therefore itdoes not make sense to give a procedure for finding somethings




Caution: Even for continuous functions, the closed interval methodworks because we have Extreme value theorem. If function is notcontinuous, then EVT cannot be invoked, as a consequence, we donot know the existence of extreme values for the function. Therefore itdoes not make sense to give a procedure for finding somethings whichdo not exist.


Problems

True/False1 If f (c) = 0, the f has a local maximum or minimum at c.


Problems

True/False1 If f (c) = 0, the f has a local maximum or minimum at c.2 If f is continuous on (a,b), then f attains an absolute maximum

value f (c) and an absolute minimum value f (d) at some numbersc,d in (a,b).


Problems



1 Sketch the graph of a function that has a local maximum at 2 andis differentiable at 2.


Problems




2 Sketch the graph of a function that has three local minima, twolocal maxima, and seven critical numbers.


Problems





3 Sketch the graphs of a function on [1,2] that is discontinuous buthas both an absolute maximum and an absolute minimum.


Problems






4 Does f (x) = ex has a local minimum?


Problems






4 Does f (x) = ex has a local minimum? local maximum?


Problems






4 Does f (x) = ex has a local minimum? local maximum? Can yougeneralise this?


SLecture4 Maxima Minima

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