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Question: Given the Least Upper Bound Property (definition 1.1.3 below), Proposition 2.1.10 below, the Bolzano-Weierstrass Theorem (Theorem 2.3.8 below), the Min-Ma...

Question

Question: Given the Least Upper Bound Property (definition 1.1.3 below), Proposition 2.1.10 below, the Bolzano-Weierstrass Theorem (Theorem 2.3.8 below), the Min-Max Theorem (Theorem 3.3.2 below), Definition 4.2.1, Lemma 4.2.2,and Rolle's Theorem (Theorem 4.2.3 below). Describe how these are connected to obtain complete proof of the Mean Value Theorem (Theorem 4.2.4 below) and prove the theorem_ Note will give thumbs up for cetailed answer. can also breakthese into separate questions necess

Question: Given the Least Upper Bound Property (definition 1.1.3 below), Proposition 2.1.10 below, the Bolzano-Weierstrass Theorem (Theorem 2.3.8 below), the Min-Max Theorem (Theorem 3.3.2 below), Definition 4.2.1, Lemma 4.2.2,and Rolle's Theorem (Theorem 4.2.3 below). Describe how these are connected to obtain complete proof of the Mean Value Theorem (Theorem 4.2.4 below) and prove the theorem_ Note will give thumbs up for cetailed answer. can also breakthese into separate questions necessany. Delinition An ordend sel has the least-upper-bound propenty if every nonempty subset that is bounded aboie has least upper bound, that SupP exists in $Froposition LIO. miondrole SCGMece {xa} bounded if and only if it convergen: Furuermon i {X, onafon increasing bounded. Ihen sup{Xa eN} If {saa} montolone decreasing (d bounded then lim inf{Xn : n N} Theorem 2}.8 (Bolzano_Weierstrass . Suppose sequence {xn} of real numbers bounded Then there exists convergent subsequence {Xrs} Theorem 3 .2 (Minimum-maximum theorem) Let f: [0.b] Rbe continuous function Then achieves both an absolute minit (ld &n absolute Paxirm on [a.b]: Delinition +1. Let$ Rb a sel and let f : Rbe a funelion. The function is said MaVd 4relative #IcLLIDHANI at â‚¬ = if thete exists = 0 such that for allx where have flx) < flc)- The definition of relative minimum AmToCOl Lemna 422 Lct f:l4.b] utriclon diferemtiable &l â‚¬ (4.b). ard relulut naanneelee reltive maximum Ut â‚¬: Then Theonm 42} (Rolle). Let f : [a.b] + Rbe cOntimuons function differentiable on (4.b) Such ahat f(a) f(6). Then there exists & â‚¬ (a.b) such thar f"(c) Theorem 424 (Mean value theorem). Let f : lu.b] Rbe FOHIomNuuclon diferentiable (0,b} Then there exists Fau GCWc f() f(e) f"(cl(b

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