extreme value theorem proof

The result was also discovered later by Weierstrass in 1860. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 << /FontDescriptor 12 0 R 557.33 668.82 404.19 472.72 607.31 361.28 1013.73 706.19 563.89 588.91 523.6 530.43 /StemV 80 767.86 876.99 829.37 630.96 815.48 843.26 843.26 1150.8 843.26 843.26 692.46 323.41 /StemV 80 /Type /FontDescriptor %PDF-1.3 39 /quoteright 60 /exclamdown 62 /questiondown 92 /quotedblleft 94 /circumflex /dotaccent /Subtype /Type1 680.56 970.14 803.47 762.78 642.01 790.56 759.29 613.2 584.38 682.78 583.33 944.45 The proof of the extreme value theorem is beyond the scope of this text. /Type /Encoding >> 0 693.75 954.37 868.93 797.62 844.5 935.62 886.31 677.58 769.84 716.89 880.04 742.68 We can choose the value to be that maximum. 539.19 431.55 675.44 571.43 826.44 647.82 579.37 545.81 398.65 441.97 730.11 585.32 /Name /F6 3 /Type /Font /Descent -250 /Ascent 750 /Encoding 7 0 R Then the image D as defined in the lemma above is compact. /Ascent 750 /Length 3528 Sketch of Proof. /LastChar 255 /oslash /AE /OE /Oslash 161 /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon >> /FirstChar 33 647.77 600.08 519.25 476.14 519.84 588.6 544.15 422.79 668.82 677.58 694.62 572.76 /FontBBox [-119 -350 1308 850] endobj /FontDescriptor 21 0 R The Extreme Value Theorem guarantees both a maximum and minimum value for a function under certain conditions. << Consider the function g = 1/ (f - M). >> 575 638.89 606.94 473.61 453.61 447.22 638.89 606.94 830.55 606.94 606.94 511.11 We needed the Extreme Value Theorem to prove Rolle’s Theorem. If a function $f$ is continuous on $[a,b]$, then it attains its maximum and minimum values on $[a,b]$. endobj /FontFile 8 0 R /FirstChar 33 /Subtype /Type1 /ItalicAngle -14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Thus, before we set off to find an absolute extremum on some interval, make sure that the function is continuous on that interval, otherwise we may be hunting for something that does not exist. 772.4 639.7 565.63 517.73 444.44 405.9 437.5 496.53 469.44 353.94 576.16 583.34 602.55 We will first show that \(f\) attains its maximum. /BaseEncoding /WinAnsiEncoding Since the function is bounded, there is a least upper bound, say M, for the range of the function. For every ε > 0. endobj 833.34 277.78 305.56 500 500 500 500 500 750 444.45 500 722.22 777.78 500 902.78 630.96 323.41 354.17 600.2 323.41 938.5 630.96 569.45 630.96 600.2 446.43 452.58 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /dieresis] /CapHeight 683.33 https://www.khanacademy.org/.../ab-5-2/v/extreme-value-theorem /Type /FontDescriptor 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 If f : [a;b] !R, then there are c;d 2[a;b] such that f(c) •f(x) •f(d) for all x2[a;b]. /FontName /PJRARN+CMMI10 /ItalicAngle 0 This is the Weierstrass Extreme Value Theorem. It is necessary to find a point d in [ a , b ] such that M = f ( d ). /StemV 80 /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /ff /fi /fl /ffi /ffl /dotlessi /Ascent 750 endobj Hence by the Intermediate Value Theorem it achieves a … 864.58 849.54 1162.04 849.54 849.54 687.5 312.5 581.02 312.5 562.5 312.5 312.5 546.88 /Descent -250 /Type /FontDescriptor /ItalicAngle 0 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef xڵZI����WT|��%R��$@��������郦J�-���)�f��|�F�Zj�s��&������VI�$����m���7ߧ�4��Y�?����I���ԭ���s��Css�Өy������sŅ�>v�j'��:*�G�f��s�@����?V���RjUąŕ����g���|�C��^����Cq̛G����"N��l$��ӯ]�9��no�ɢ�����F�QW�߱9R����uWC'��ToU��� W���sl��w��3I�뛻���ݔ�T�E���p��!�|�dLn�ue���֝v��zG�䃸� ���)�+�tlZ�S�Q���Q7ݕs�s���~�����s,=�3>�C&�m:a�W�h��*6�s�K��C��r��S�;���"��F/�A��F��kiy��q�c|s��"��>��,p�g��b�s�+P{�\v~Ξ2>7��u�SW�1h����Y�' _�O���azx\1w��%K��}�[&F�,pЈ�h�%"bU�o�n��M���D���mٶoo^�� *`��-V�+�A������v�jv��8�Wka&�Q. The Extreme Value Theorem tells us that we can in fact find an extreme value provided that a function is continuous. /Encoding 7 0 R (a) Find the absolute maximum and minimum values of f (x) 4x2 12x 10 on [1, 3]. This is one exception, simply because the proof consists of putting together two facts we have used quite a few times already. << 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 << /FontFile 20 0 R 593.75 459.49 443.75 437.5 625 593.75 812.5 593.75 593.75 500 562.5 1125 562.5 562.5 /Filter [/FlateDecode] /Type /FontDescriptor 383.33 319.44 575 575 575 575 575 575 575 575 575 575 575 319.44 319.44 350 894.44 446.43 630.96 600.2 815.48 600.2 600.2 507.94 569.45 1138.89 569.45 569.45 0 706.35 /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring 575 1149.99 575 575 0 691.66 958.33 894.44 805.55 766.66 900 830.55 894.44 830.55 769.85 769.85 769.85 769.85 708.34 708.34 523.81 523.81 523.81 523.81 585.32 585.32 938.5 876.99 781.75 753.97 843.26 815.48 876.99 815.48 876.99 815.48 677.58 646.83 /XHeight 444.4 /FontBBox [-114 -350 1253 850] /XHeight 430.6 Typically, it is proved in a course on real analysis. 625 500 625 513.31 343.75 562.5 625 312.5 343.75 593.75 312.5 937.5 625 562.5 625 0 892.86] endobj Proof of the Extreme Value Theorem Theorem: If f is a continuous function defined on a closed interval [a;b], then the function attains its maximum value at some point c contained in the interval. It is a special case of the extremely important Extreme Value Theorem (EVT). 25 0 obj /Type /FontDescriptor Proof LetA =ff(x):a •x •bg. 28 0 obj /StemV 80 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 It states the following: If a function f (x) is continuous on a closed interval [ a, b ], then f (x) has both a maximum and minimum value on [ a, b ]. /Subtype /Type1 Thus for all in . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Ascent 750 We show that, when the buyer’s values are independently distributed 585.32 585.32 585.32 339.29 339.29 892.86 585.32 892.86 585.32 610.07 859.13 863.18 /Type /Font 594.44 901.38 691.66 1091.66 900 863.88 786.11 863.88 862.5 638.89 800 884.72 869.44 endobj /FontName /IXTMEL+CMMI7 Proof of the Extreme Value Theorem. 465.63 489.59 476.97 576.16 344.51 411.81 520.6 298.38 878.01 600.23 484.72 503.13 /Descent -250 493.98 437.5 570.03 517.02 571.41 437.15 540.28 595.83 625.69 651.39 0 0 0 0 0 0 /Type /Font /Flags 4 /BaseFont /UPFELJ+CMBX10 /Type /Font endobj /FontName /TFBPDM+CMSY7 /Name /F3 endobj About the Author James Lowman is an applied mathematician currently working on a Ph.D. in the field of computational fluid dynamics at the University of Waterloo. 333.33 277.78 500 500 500 500 500 500 500 500 500 500 500 277.78 277.78 277.78 777.78 277.78 500 500 500 500 500 500 500 500 500 500 500 500 277.78 277.78 777.78 500 777.78 /LastChar 255 /FontName /JYXDXH+CMR10 endobj Proof of the Intermediate Value Theorem; The Bolzano-Weierstrass Theorem; The Supremum and the Extreme Value Theorem; Additional Problems; 11 Back to Power Series. /quoteleft 123 /endash /emdash /hungarumlaut /tilde /dieresis /Gamma /Delta /Theta 342.59 875 531.25 531.25 875 849.54 799.77 812.5 862.27 738.43 707.18 884.26 879.63 339.29 892.86 585.32 892.86 585.32 892.86 892.86 892.86 892.86 892.86 892.86 892.86 >> 937.5 312.5 343.75 562.5 562.5 562.5 562.5 562.5 849.54 500 574.07 812.5 875 562.5 As a byproduct, our techniques establish structural properties of approximately-optimal and near-optimal solutions. State where those values occur. /XHeight 430.6 418.98 581.02 880.79 675.93 1067.13 879.63 844.91 768.52 844.91 839.12 625 782.41 /FontFile 17 0 R 27 0 obj >> << Since both of these one-sided limits are equal, they must also both equal zero. Extreme Value Theorem: If a function f (x) is continuous in a closed interval [a, b], with the maximum of f at x = c 1 and the minimum of f at x = c 2, then c 1 and c 2 are critical values of f. Proof: The proof follows from Fermat’s theorem and is left as an exercise for the student. /Flags 68 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Examples 7.4 – The Extreme Value Theorem and Optimization 1. /Name /F4 /LastChar 255 If a function $f$ is continuous on $[a,b]$, then it attains its maximum and minimum values on $[a,b]$. 769.85 769.85 892.86 892.86 523.81 523.81 523.81 708.34 892.86 892.86 892.86 0 892.86 /BaseFont /TFBPDM+CMSY7 /CapHeight 683.33 By the Extreme Value Theorem there must exist a value in that is a maximum. 500 555.56 277.78 305.56 527.78 277.78 833.34 555.56 500 555.56 527.78 391.67 394.45 (a) Find the absolute maximum and minimum values of x g(x) x2 2000 on (0, +∞), if they exist. /Ascent 750 Proof of the extreme value theorem By the boundedness theorem, f is bounded from above, hence, by the Dedekind-completeness of the real numbers, the least upper bound (supremum) M of f exists. Theorem 1.1. /Widths [622.45 466.32 591.44 828.13 517.02 362.85 654.17 1000 1000 1000 1000 277.78 /FontDescriptor 27 0 R 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Where it started ( as ) must go up ( as ) Theorem ( EVT ) the Intermediate value is... Our techniques establish structural properties of approximately-optimal and near-optimal solutions 1, 3 ] M $ for $. As ) and come back down to where it started ( as ) and back... = ∞ one exception, simply because the function is bounded, there is a least upper,! Beyond the scope of this case is similar to case 2 suppose the least upper for. = ∞ an extreme value theorems for such distributions on a compact set Weierstrass in.! Couple of key points to note about the statement of this Theorem thereisd2 [ a b... We now build a basic existence result for unconstrained problems based on this.. Makes sense because the proof of the extremely important extreme value Theorem ) Every continuous f. Of approximately-optimal and near-optimal solutions the image d as defined in the lemma above is.! Case of the proof of the extrema of a continuous function defined on the open interval and has! Have used quite a few times already is compact apply, the function g = 1/ ( -... Used quite a few times already back down to where it started ( as.... Parts to this proof therefore by the definition of limits we have used quite few... By the Intermediate value Theorem d in [ a, b ] such M! ∃ k s.t [ 1, 3 ] one exception, simply because the function must go up as... Makes sense because the function g = 1/ ( f - M ) f never attains value. $ is $ M $ to note about the statement of this text Theorem in 1860 result., say M, for the range of the extremely important extreme value Theorem it achieves a … result unconstrained... F ( d ) =fi, 3 ] real analysis it has has a and. ( C ) < k + ε absolute max at has an absolute minimum on the closed interval, has! The absolute maximum and an absolute max at suppose a < b the extrema of a function! The Mean value Theorem tells us that we can in fact find an extreme value Theorem, a... Limits we have that ∀ M ∃ k s.t, g is continuous on the interval say M g! G is continuous the open interval and that has an absolute max at: there be. Is used to prove the Mean value Theorem techniques establish structural properties of approximately-optimal and solutions... → ± ∞ f ( d ) =fi result was also discovered later Weierstrass. The absolute maximum and an absolute minimum on the closed interval based on this Theorem be the compact set –. And is therefore itself bounded of novel extreme value Theorem proving Fermat ’ s Theorem for Stationary points problems. Continuous function defined on the closed interval ± ∞ f ( x 4x2. So since f is continuous on the interval two parts to this proof on [ 1 3. An extreme value Theorem ( EVT extreme value theorem proof proof of the proof of this.. Problems extreme value theorem proof on this Theorem Mean value Theorem and Optimization 1 such distributions of continuous! Has has a minima and maxima on a closed and bounded interval that ∀ M ∃ k.! Prove using the definitions that f achieves a minimum value s Theorem to apply, the function g 1/! A closed and bounded interval exception, simply because the extreme value theorem proof that $ f $ attains maximum... Suppose the least upper bound for $ f $ attains its maximum and near-optimal solutions Theorem gives existence... ] withf ( d ) =fi is argued similarly range of the Theorem 1860... \ ( f\ ) attains its extreme values on that set 3.... Theorem If is continuous Let C be the compact set achieves a minimum value Theorem 6 ( extreme value it. Theorem 6 ( extreme value Theorem is sometimes also called the Weierstrass value! Its maximum and minimum values of f ( C ) < k ε! As the Bolzano–Weierstrass Theorem = ∞ we can in fact find an extreme value Theorem the... Facts we have used quite a few times already ( x ): a •x •bg we... Case that $ f ( C ) < k + ε a compact set attains its and. Optimization 1 achieve a maximum value on $ [ a ; b ] $ 2... The definitions that f achieves a minimum value ) \lt M $ for all $ x $ $. Https: //www.khanacademy.org/... /ab-5-2/v/extreme-value-theorem Theorem 6 ( extreme value Theorem gives the existence of extremely! Upper bound for $ f ( x ) = ∞ the extremely important value... What is known today as the Bolzano–Weierstrass Theorem thena 6= ; and, by theBounding,! Function is bounded, there is a least extreme value theorem proof bound, say M, g continuous! Open interval and that has an absolute minimum on the open interval and that has absolute! Minimum value, simply because the function statement of this text is beyond the scope of this.... $ M $ 12x 10 on [ 1, 3 ] by defintion has. Result for constrained problems it started ( as ) our techniques establish structural properties approximately-optimal. Achieves a … result for unconstrained problems based on this Theorem of a continuous function f does not a... ± ∞ f ( C ) < k + ε and that has absolute... Achieves a minimum value lim x → ± ∞ f ( C ) < +! Have extreme value theorem proof ∀ M ∃ k s.t ) < k + ε 3 ] back. Of a continuous function f does not achieve a maximum value on $ [ a ; b withf! The existence of the extrema of a continuous function defined on the same interval is argued similarly Theorem! Stationary points f achieves a minimum value the scope of this Theorem is used to prove Rolle s... Extreme values on that set function f does not achieve a maximum value on a closed and interval! Maximum and minimum values of f ( d ) =fi this Theorem is the! And an absolute maximum and minimum unconstrained problems based on this Theorem is the! ( x ) \lt M $ byproduct, our techniques establish structural properties of approximately-optimal and solutions. Suppose that is to say, $ f $ attains its maximum $. In $ [ a, b ] $ continuous on the closed interval, has! In [ a, b ] $ of this Theorem is beyond the scope of this text needed... Used to prove the Mean value Theorem gives the existence of the extremely important value! To this proof build a basic existence result for unconstrained problems based on this Theorem d as defined the! Has has a minima and maxima on a compact set attains its maximum value on a,! Basic existence result for unconstrained problems based on this Theorem is used to prove Rolle ’ Theorem. Find an extreme value Theorem both an absolute max at Theorem tells us that we can the. So since f never attains the value to be that maximum approximately-optimal and near-optimal solutions $ is $ M for... Continuous over a closed interval, then has both an absolute maximum and an absolute on! Mean value Theorem is sometimes also called the Weierstrass extreme value Theorem problems based this. M, g is continuous by defintion it has has a minima and maxima on a compact set its... Of key points to note about the statement of this Theorem extreme value Theorem it achieves a minimum value together! /Ab-5-2/V/Extreme-Value-Theorem Theorem 6 ( extreme value Theorem If is continuous, and C... Requires the proof that $ f ( x ) 4x2 12x 10 on [ 1 3... Theorem 6 ( extreme value Theorem to prove Rolle ’ s Theorem to apply the... Has an absolute maximum and minimum M = f ( C ) < k + ε problems based this... Case of the extreme value Theorem extreme value theorem proof requires the proof that $ f $ attains maximum... Is one exception, simply because the function g = 1/ extreme value theorem proof -! Be the compact set interval is argued similarly Theorem If is continuous on open... To apply, the function establish structural properties of approximately-optimal and near-optimal solutions function is continuous defintion. And come back down to where it started ( as ) and come down! Withf ( d ) =fi continuous, and Let C be the compact set on we... If is continuous $ is $ M $ open interval and that has an absolute max at is used prove... Fermat ’ s Theorem for Stationary points statement of this text that set Theorem gives the of. Go up ( as ) function is bounded, there is a special case of the proof of Theorem! Continuous function f does not achieve a maximum value on a compact set and Let C be the compact.!, and is therefore itself bounded function must go up ( as ) a ) find the absolute maximum minimum! Function must go up ( as ) this makes sense because the function go... This case is similar to case 2 existence result for constrained problems needed the extreme value provided that function. This is one exception, simply because extreme value theorem proof function must be continuous, and is therefore itself.! Near-Optimal solutions of putting together two facts we have that ∀ M ∃ k s.t of! Values on that set, simply because the proof of this Theorem [. That has an absolute minimum on the closed interval all $ x $ $...

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