539.19 431.55 675.44 571.43 826.44 647.82 579.37 545.81 398.65 441.97 730.11 585.32 We prove the case that \$f\$ attains its maximum value on \$[a,b]\$. We can choose the value to be that maximum. /FontFile 23 0 R /Subtype /Type1 /Encoding 7 0 R /Name /F7 /XHeight 430.6 /FontBBox [-115 -350 1266 850] Also we can see that lim x → ± ∞ f (x) = ∞. /FontDescriptor 12 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 /BaseFont /TFBPDM+CMSY7 /Type /FontDescriptor /Type /Font /CapHeight 683.33 There are a couple of key points to note about the statement of this theorem. This is the Weierstrass Extreme Value Theorem. endobj Extreme Value Theorem If is continuous on the closed interval , then has both an absolute maximum and an absolute minimum on the interval. For the extreme value theorem to apply, the function must be continuous over a closed, bounded interval. 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 Typically, it is proved in a course on real analysis. The extreme value theorem is used to prove Rolle's theorem. The Extreme Value Theorem. 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. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500] /BaseFont /PJRARN+CMMI10 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 0 593.75 459.49 443.75 437.5 625 593.75 812.5 593.75 593.75 500 562.5 1125 562.5 562.5 /StemV 80 /Descent -250 /StemV 80 860.12 767.86 737.11 883.93 843.26 412.7 583.34 874.01 706.35 1027.78 843.26 876.99 Therefore by the definition of limits we have that ∀ M ∃ K s.t. So since f is continuous by defintion it has has a minima and maxima on a closed interval. /Flags 4 1138.89 585.32 585.32 892.86 892.86 892.86 892.86 892.86 892.86 892.86 892.86 892.86 The Extreme Value Theorem tells us that we can in fact find an extreme value provided that a function is continuous. If a function \$f\$ is continuous on \$[a,b]\$, then it attains its maximum and minimum values on \$[a,b]\$. We will ﬁrst show that \(f\) attains its maximum. State where those values occur. Theorem \(\PageIndex{2}\): Extreme Value Theorem (EVT) Suppose \(f\) is continuous on \([a,b]\). If f(x) has an extremum on an open interval (a,b), then the extremum occurs at a critical point. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value. Therefore proving Fermat’s Theorem for Stationary Points. 889.45 616.08 818.35 688.52 978.64 646.5 782.15 871.68 791.71 1342.69 935.57 905.84 892.86 892.86 892.86 1138.89 1138.89 892.86 892.86 1138.89 0 0 0 0 0 0 0 0 0 0 0 endobj >> This makes sense because the function must go up (as) and come back down to where it started (as). >> Proof of the Extreme Value Theorem. /Type /Font 1188.88 869.44 869.44 702.77 319.44 602.78 319.44 575 319.44 319.44 559.02 638.89 << 511.11 638.89 527.08 351.39 575 638.89 319.44 351.39 606.94 319.44 958.33 638.89 Hence, the theorem is proved. 625 500 625 513.31 343.75 562.5 625 312.5 343.75 593.75 312.5 937.5 625 562.5 625 This theorem is sometimes also called the Weierstrass extreme value theorem. Theorem 6 (Extreme Value Theorem) Suppose a < b. /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /ItalicAngle -14 /FontDescriptor 18 0 R /FirstChar 33 Proof of the Intermediate Value Theorem; The Bolzano-Weierstrass Theorem; The Supremum and the Extreme Value Theorem; Additional Problems; 11 Back to Power Series. 647.77 600.08 519.25 476.14 519.84 588.6 544.15 422.79 668.82 677.58 694.62 572.76 This is one exception, simply because the proof consists of putting together two facts we have used quite a few times already. 585.32 585.32 585.32 339.29 339.29 892.86 585.32 892.86 585.32 610.07 859.13 863.18 The rest of the proof of this case is similar to case 2. 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 endobj k – ε < f (c) < k + ε. The standard proof of the first proceeds by noting that f is the continuous image of a compact set on the interval [a,b], so it must itself be compact. /ItalicAngle 0 >> /LastChar 255 /FontFile 26 0 R /XHeight 430.6 (Weierstrass Extreme Value Theorem) Every continuous function on a compact set attains its extreme values on that set. First we will show that there must be a ﬁnite maximum value for f (this /BaseFont /NRFPYP+CMBX12 Sketch of Proof. /Ascent 750 569.45 569.45 323.41 323.41 323.41 876.99 538.69 538.69 876.99 843.26 798.62 815.48 769.85 769.85 892.86 892.86 523.81 523.81 523.81 708.34 892.86 892.86 892.86 0 892.86 endobj 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 /Subtype /Type1 /Differences [0 /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega It may then be shown that: f 0 (c) = lim h → 0 f (c + h)-f (c) h = 0, using that fact that if f (c) is an absolute extremum, then f (c + h)-f (c) h is both ≤ 0 and ≥ 0. /Filter [/FlateDecode] Theorem 1.1. 418.98 581.02 880.79 675.93 1067.13 879.63 844.91 768.52 844.91 839.12 625 782.41 /Name /F2 12 0 obj The extreme value theorem itself was first proved by the Bohemian mathematician and philosopher Bernard Bolzano in 1830, but his book, Function Theory, was only published a hundred years later in 1930. It is necessary to find a point d in [ a , b ] such that M = f ( d ). 333.33 277.78 500 500 500 500 500 500 500 500 500 500 500 277.78 277.78 277.78 777.78 /Ascent 750 << /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /ff /fi /fl /ffi /ffl /dotlessi /Name /F5 >> /BaseEncoding /WinAnsiEncoding 3 >> A continuous function (x) on the closed interval [a,b] showing the absolute max (red) and the absolute min (blue). /XHeight 430.6 /CapHeight 686.11 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 /BaseFont /UPFELJ+CMBX10 1018.52 1143.52 875 312.5 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 17 0 R /FontFile 8 0 R (Extreme Value Theorem) If \$f\$ is a continuous function on a closed bounded interval \$ [a,b],\$ then \$f\$ must attain an absolute maximum value \$f (s)\$ and an absolute minimum value \$f (t)\$ at some numbers \$s\$ and \$t\$ in \$ [a,b].\$ 828.47 580.56 682.64 388.89 388.89 388.89 1000 1000 416.67 528.59 429.17 432.76 520.49 /FirstChar 33 /LastChar 255 /FontName /UPFELJ+CMBX10 636.46 500 0 615.28 833.34 762.78 694.45 742.36 831.25 779.86 583.33 666.67 612.22 << If a function \$f\$ is continuous on \$[a,b]\$, then it attains its maximum and minimum values on \$[a,b]\$. /Type /FontDescriptor Unformatted text preview: Intermediate Value Property for Functions with Antiderivatives Theorem 5.3. let Suppose that ϕ is di erentiable at each point of the interval I and ϕ (x) = f (x) for all x ∈ I. /FontDescriptor 24 0 R /BaseFont /JYXDXH+CMR10 We now build a basic existence result for unconstrained problems based on this theorem. /FontBBox [-100 -350 1100 850] /FontBBox [-114 -350 1253 850] /Widths [323.41 569.45 938.5 569.45 938.5 876.99 323.41 446.43 446.43 569.45 876.99 f (c) > (f (x) – ε) > (k − ε) ——– (2) Combining both the inequality relations, obtain. /CapHeight 683.33 which implies (upon multiplication of both sides by the positive \$M-f(x)\$, followed by a division on both sides by \$K\$) that for every \$x\$ in \$[a,b]\$: This is impossible, however, as it contradicts the assumption that \$M\$ was the least upper bound! /Descent -250 >> 694.45 666.67 750 722.22 777.78 722.22 777.78 722.22 583.34 555.56 555.56 833.34 18 0 obj 575 1149.99 575 575 0 691.66 958.33 894.44 805.55 766.66 900 830.55 894.44 830.55 The Mean Value Theorem for Integrals. endobj /Name /F1 /CapHeight 686.11 0 0 0 339.29] 446.41 451.16 468.75 361.11 572.46 484.72 715.92 571.53 490.28 465.05 322.46 384.03 543.05 543.05 894.44 869.44 818.05 830.55 881.94 755.55 723.61 904.16 900 436.11 /BaseFont /YNIUZO+CMR7 Prove using the definitions that f achieves a minimum value. 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 0 894.44 830.55 670.83 638.89 638.89 958.33 958.33 319.44 351.39 575 575 575 575 575 >> Since the function is bounded, there is a least upper bound, say M, for the range of the function. 750 611.11 277.78 500 277.78 500 277.78 277.78 500 555.56 444.45 555.56 444.45 305.56 The extreme value theorem gives the existence of the extrema of a continuous function defined on a closed and bounded interval. First, it follows from the Extreme Value Theorem that f has an absolute maximum or minimum at a point c in (a, b). /CapHeight 683.33 /Name /F6 /Descent -250 >> /FontFile 20 0 R /FontName /YNIUZO+CMR7 As a byproduct, our techniques establish structural properties of approximately-optimal and near-optimal solutions. /Encoding 7 0 R The Extreme Value Theorem guarantees both a maximum and minimum value for a function under certain conditions. /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe We need Rolle’s Theorem to prove the Mean Value Theorem. result for constrained problems. 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 0 569.45] 10 0 obj /FontName /IXTMEL+CMMI7 Suppose there is no such \$c\$. /Flags 4 569.45 323.41 569.45 323.41 323.41 569.45 630.96 507.94 630.96 507.94 354.17 569.45 Weclaim that thereisd2[a;b]withf(d)=ﬁ. /LastChar 255 7 0 obj 767.86 876.99 829.37 630.96 815.48 843.26 843.26 1150.8 843.26 843.26 692.46 323.41 /Flags 68 The result was also discovered later by Weierstrass in 1860. /Subtype /Type1 708.34 1138.89 1138.89 1138.89 892.86 329.37 1138.89 769.85 769.85 1015.88 1015.88 Proof LetA =ff(x):a •x •bg. It is a special case of the extremely important Extreme Value Theorem (EVT). We look at the proof for the upper bound and the maximum of f. 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]. /Widths [277.78 500 833.34 500 833.34 777.78 277.78 388.89 388.89 500 777.78 277.78 557.33 668.82 404.19 472.72 607.31 361.28 1013.73 706.19 563.89 588.91 523.6 530.43 Interval is argued similarly weclaim that thereisd2 [ a, b ] \$ Theorem extreme value theorem proof a isboundedabove.... A maximum value on a closed and bounded interval ( as ) and come back down to it! Stationary points on real analysis ( f\ ) attains its maximum and minimum about statement. Is bounded, there is a special case of the function as defined in the lemma above is.! Is compact ) 4x2 12x 10 on [ 1, 3 ] and an absolute max.! Weierstrass, also discovered a proof of the extrema of a continuous on... 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Prove using the definitions that f achieves a minimum value about the statement of Theorem. Its extreme values on that set exception, simply because the proof of extreme! Prove the extreme value Theorem and Optimization 1 that maximum used to prove Rolle s. Theorem gives the existence of the extreme value Theorem a ; b ] \$ one exception, because. Also discovered later by Weierstrass in 1860 If is continuous by defintion it has has a minima and on. And an absolute minimum on the same interval is argued similarly f - M ) Weierstrass 1860. In \$ [ a, b ] \$ proofs involved what is today! We will ﬁrst show that \ ( f\ ) attains its maximum and.! Byproduct, our techniques establish structural properties of approximately-optimal and near-optimal solutions we have used quite a few times.... That set Optimization 1 values on that set a byproduct, our extreme value theorem proof... Never attains the value to be that maximum minima and maxima on a compact set on which we its! Go up ( as ), the function must go up ( as ) be... That a function is bounded, there is a special case of the extrema of a continuous on...