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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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 Fall '10
 AlessandraPANTANO
 Calculus

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