extremes+2+ - Maximum and Minimum Values sec/on 4.1...

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Unformatted text preview: Maximum and Minimum Values sec/on 4.1 Extreme Values of a function (on a set ) Rela%ve Maximum Rela%ve Minimum Extreme Values of a function (on a set ) absolute (and Rela/ve) maximum Absolute Maximum rela/ve maximum Rela%ve Maximum Rela%ve Minimum rela/ve minimum absolute minimum absolute (and rela/ve) maximum rela/ve maximum rela/ve minimum absolute minimum A point x0 is called… • Absolute Maximum: if f(x0) ≥ f(x) for all x in the domain •  Absolute minimum: if f(x0) ≤ f(x) for all x in the domain •  Rela%ve Maximum: if f(x0) ≥ f(x) for all x near x0 •  Rela%ve minimum: if f(x0) ≤ f(x) for all x near x0 in an open interval around x0 Absolute Maxima & Minima Relative Maxima & Minima The value of f is bigger at a than at the points nearby The value of f is smaller at a than at the points nearby Extreme Values of a function (on a set ) global Rela%ve Maximum local Rela%ve Minimum Extreme Values of a function (on a set ) Global Maximum global maximum local maximum Local Maximum Local Minimum A B C local minimum global minimum D Example: y=cos(x) (defined over all real) The maximum value is +1. The minimum value is  ­1. Extremes: y=cos(x) takes its absolute (and rela/ve) maximum at x0 = 2n π. y=cos(x) takes its absolute (and rela/ve) minimum at: x0 = (2n+1) π. Absolute Extremes Ques2on Every func/on has an absolute maximum and an absolute minimum in its domain. a) True b) False Example: y=x2 (defined over all real) There is no maximum value. There is no highest point The minimum value is 0. Extremes: There is no absolute (nor rela/ve) maximum. There is an absolute (and rela/ve) minimum at x0 = 0. Example: y=x2 (defined over [-2,2]) This is an absolute maximum, but not a rela7ve maximum The maximum value is 4. The minimum value is 0. Extremes: There is an absolute maximum at x0 = ±2 . There is no rela7ve maximum. There is an absolute (and rela/ve) minimum at x0 = 0. Example: y=x2 (defined over [-1,2]) Ques2on The value +1 is A)  An absolute maximum B)  A rela/ve maximum C)  None of the above Example: y=x2 (defined over [-1,2]) Answer The value +1 is A)  An absolute maximum B)  A rela/ve maximum C)  None of the above It is not an absolute maximum (because 4 is a higher value). It is not a rela7ve maximum because it is achieved at an end ­point. Example: y=x3 (defined over all real) Ques2on The func/on y=x3 has A)  An absolute maximum B)  A rela/ve maximum C)  An absolute minimum D)  A rela/ve minimum E)  None of the above Example: y=x3 (defined over all real) Answer The func/on y=x3 has A)  An absolute maximum B)  A rela/ve maximum C)  An absolute minimum It has not an absolute extremes. It has no rela7ve extremes. D)  A rela/ve minimum E)  None of the above Example: y=x3 (defined over [-2,2]) Ques2on The func/on y=x3 has A)  An absolute minimum on [ ­2,2] B)  A rela/ve minimum on [ ­2,2] C)  None of the above D)  Both an absolute and a rela/ve minimum on [ ­2,2] Example: y=x3 (defined over [-2,2]) Answer The func/on y=x3 has A)  An absolute minimum on [ ­2,2] B)  A rela/ve minimum on [ ­2,2] C)  None of the above absolute minimum at c= ­2 D)  Both an absolute and a rela/ve minimum on [ ­2,2] It is not a rela7ve minimum because it is achieved at an end ­point. Another Example: Find all the rela/ve and absolute extremes Answer: There are no absolute extremes rela/ve maximum at c= ­3 rela/ve maximum at c=2 rela/ve minimum at c=4 rela/ve minimum at c= ­1 Some func%ons have absolute extremes values, others do not. abs. max. at 2 y=x2 on [ ­1,2] y=x2 on on [ ­1,0)u(0,2] abs. max. at 2 abs. min. at 0 No abs. min. The point 0 Is not included No abs. max. y=x2 on ( ­∞,+∞) abs. min. at 0 The existence of absolute extremes values depends on the func%on but also on the domain The Extreme Value Theorem =f(c) d a c b =f(d) If f is con%nuous on a closed interval [a,b], then f aDains an absolute maximum value f(c) and an absolute minimum value f(d) at some points c and d in [a,b]. The Extreme Value Theorem d c c d d c Absolute extremes may occur at the end ­points! The Extreme Value Theorem d1 c1 c2 d2 Absolute extremes may occur more than once! The Extreme Value Theorem The point c Is not included If f is con%nuous on [a,b], then f has an absolute minimum and and absolute maximum No Abs Min. This func/on does not violate the Extreme Value Theorem because A)  f is not con/nuous on its domain B)  The domain is not a closed interval The Extreme Value Theorem The point c Is not included If f is con%nuous on [a,b], then f has an absolute minimum and and absolute maximum No Abs Min. This func/on does not violate the Extreme Value Theorem because A)  f is not con/nuous on its domain B)  The domain is not a closed interval f is con7nuous on its domain! The Extreme Value Theorem If f is con%nuous on [a,b], then f has an absolute minimum and and absolute maximum No Abs Min. This func/on does not violate the Extreme Value Theorem because A)  f is not con/nuous on its domain B)  The domain is not a closed interval The Extreme Value Theorem If f is con%nuous on [a,b], then f has an absolute minimum and and absolute maximum No Abs Min. This func/on does not violate the Extreme Value Theorem because A)  f is not con/nuous on its domain B)  The domain is not a closed interval The domain of f is all of [a,b]! Local Extremes Ques2on If c is a local maximum or a local minimum for f, then the tangent line at c is horizontal. a) True b) False Local Extremes Answer If c is a local maximum or a local minimum for f, then the tangent line at c is horizontal. a) True b) False y=|x| has a local minimum at x=0, but the tangent line at this point is not horizontal Fermat’s Theorem (on local extremes) Local maximum: horizontal tangent Local minimum: horizontal tangent If f has a local maximum or a local minimum at c, and if f ’(c) exists, then f ’(c)=0 Fermat’s Theorem This does not violate Fermat’s theorem! f ’ exists and is =0 If f has a local extreme at c and if f ’(c) exists then f ’(c)=0 f ’ does not exist f ’ exists and is =0 f ’ exists and is =0 f ’ exists and is =0 If f has a local extreme at c then either f ’(c)=0 or f ’(c) does not exist no tangent line Critical Points A point c in the domain is called a cri/cal point for f if either f ’(c)=0 or f ’(c) does not exist. Fermat’s theorem Local extremes occur at cri/cal points. Solving the equa7on f ’(c)=0 may not give you all the local extremes. Example f ’ exists and is =0 f has 2 local extremes. f ’ does not exist Only one of them can be found by se_ng f ’(c)=0. Example Ques2on The number of cri/cal points of f is equal to A)  1 B)  3 C)  5 Example Answer The number of cri/cal points of f is equal to A)  1 B)  3 C)  5 There are 3 cri7cal points (and 3 local extremes). Only one of them can be found solving the equa7on f ’(c)=0. At the remaining two, the deriva7ve does not exist. Finding local extremes Step 1) Find f ’(x) Step 2) Find all cri/cal points c • The points c where f ’ does not exist • The points c where f ’ is zero Step 3) For each cri/cal point c, understand if c is a local maximum or a local minimum LOCAL MAXIMUM LOCAL MINIMUM Finding local extremes Ques2on A local extreme can occur At an end ­point. a) True b) False LOCAL MAXIMUM LOCAL MINIMUM Finding local extremes Answer A local extreme can occur at an end ­point. a) True b) False LOCAL MAXIMUM When looking for local extremes, you don’t need to worry about end ­points. LOCAL MINIMUM Finding absolute extremes on [a,b] An absolute extreme can occur at an end ­point and at cri/cal points. When looking for absolute extremes, you need to worry about both cri7cal points and end ­points. Finding ABSOLUTE extremes on [a,b] Step 0) Compute f(a) and f(b) Step 1) Find f ’(x) Step 2) Find all cri/cal points c • The points c where f ’ does not exist • The points c where f ’ is zero Step 3) For each cri/cal point c, compute f(c) Step 4) Compare f(a) and f(b) with f(c) for every cri/cal point c, to find The biggest and the lowest value… ...
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