Question

Logic

(a) Let f : [a, b] → R be a continuous function. Prover that there exists ce [a, b] such that con la silany - be a criminatoria per le sue in elan 5(e) = So gladde

Answer 1

bet t: [a,b] tt be a Continuous function. bove that there exist Co[a,b] such that f(1) = s fcanda broof: Theorems Ibo o folo If Ů fo[a, b] =Rf go [a b] SR are both integrable on [a, b] 4 ii) gras has the Same sign to G[a,b ] then there is a number M such that M L M L M where m= Inf & f ( ) [(xe[a,b] M = Supę for I RE[a, b] 3 f S flagg og d2 = M L goy dx If for these fis Continuous on [a, b ] then there Exist a point § 6 [ab] such that ресс, асэл Ая - Scs) сасы аз Boof (of the theorem) lose I. Let glas yo tx € [a,b] Since me floy EM 4x6 [a, b] We have m gley < floog glog s m gloy ta Głam f, g are integrable on [aph ] song, fg and mg are all integrable on [ars] t im Så cas da L So stay grayda < M L granda

let gres Lo xe [a,b], en masco LM & E[ado] Mgcas fcoy glas cm glas ta tla, mdogcayda asoslang (as da Lim ograma a Combining (i) & (iig it ballows that the integral cangcandy lies between * Сас») 4x fм с 4 4 8ty feoff, same in [a, b] Therefore we can write fexy gay dx = Mgrayda Where is a real number satisfying m LMLM. . Now If fis Continuous on carb then it will assume every value between its infineon & Supremum on [a,b] at least once in [a, b] e can Importent pofesty of Continuous functions So there Exisda point & € Cap such that M = f(2), Sos'sce gay de SAD Sg cay da Where se [a, b]. Now taking along Comes to the Proof of this problem ..

above therren taking g(x) = 1 x x & [ab] In the ( we then get then F Fle de - fly (6-a) Where CERET Conti Tsince fis continuous given] No Sowe get - fces= L S floyda Debine FOR IR by a a 4 F(a) = C vite t3 ft to fore that Fly is differentiable by Ceting the appropriate the orem, Theorem ht fo[a, b] or be Continuous on [as] f oocc, d] IR, vo [c, d] - R be two differentialle functions on [c, d] Such that u Pea]) l'ann] * v*^3 zemin It to [C] IR be a bunction defined Fly= fleidt , at [c, d] vay

then is differentiable on [c, d] f 'cas = f( (M) v'cy - Cors) ulos, to de [and] clearly by the above theorem ftet is Continuous & xt & n are all di fb esentiable so by the above theorem we can conclude that is differentiable. Now, to calculate F'cas. We have F(x) = a tet3 d by above theorem. +'cz - V1 + (32 64 ) - Vitex" by - 39