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X'0, X1,* , Xn E (a, b) are arbitrary points such that x; # x; for i # j. Fur- thermore, let P be a polynomial of degree n such that P(x;) = f(x;) for all i = 0, 1, · .. ,n. Prove that for every x E (a, b) there exists ( such that min{x, xo, x1, .. , Xn} < $ < max{x, xo, x1, , xn}, and %3D f(n+1)(C) f(x) – P(x) = (x – xo)(x – x1) ·.· (x – xn). (n + 1)! | | Hint: Fix x, and denote by M the number that satisfies the equation M ;(x – xo)(x – x1) · · (x – xn). (n + 1)! f(x) – P(x) : %3D ...

X'0, X1,* , Xn E (a, b) are arbitrary points such that x; # x; for i # j. Fur- thermore, let P be a polynomial of degree n such that P(x;) = f(x;) for all i = 0, 1, · .. ,n. Prove that for every x E (a, b) there exists ( such that min{x, xo, x1, .. , Xn} < $ < max{x, xo, x1, , xn}, and %3D f(n+1)(C) f(x) – P(x) = (x – xo)(x – x1) ·.· (x – xn). (n + 1)! | | Hint: Fix x, and denote by M the number that satisfies the equation M ;(x – xo)(x – x1) · · (x – xn). (n + 1)! f(x) – P(x) : %3D ...

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 24E
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Differentiable functions 

6. Assume that f : (a,b) → R is an n +1 times differentiable function and X0, x1, · · . , Xn E (a, b) are arbitrary points such that xi + x; for i j. Fur- thermore, let P be a polynomial of degree n such that P(x;) = f(x;) for all i = 0,1, · · min{x, xo, x1, ·** .. ,n. Prove that for every x E (a, b) there exists ( such that .. , xn} < 5 < max{x, x0, x1, · f(a+1)(C) (n+ 1)! , Xn}, and f(x) – P(x) = (x – xo)(x – x1) ·. (x – xn). ... Hint: Fix x, and denote by M the number that satisfies the equation M ;(x – xo)(x – x1) · . (x – xn). (n + 1)! f(x) – P(x) = - | ... Furthermore, consider the function M g(t) = f(t) – P(t) - In i y(t – xo)(t – x1) · · · (t – xn), te (a, b), and observe that g(x) = g(x0) = g(xn) = 0. %3D =.. .- %3D
Transcribed Image Text:6. Assume that f : (a,b) → R is an n +1 times differentiable function and X0, x1, · · . , Xn E (a, b) are arbitrary points such that xi + x; for i j. Fur- thermore, let P be a polynomial of degree n such that P(x;) = f(x;) for all i = 0,1, · · min{x, xo, x1, ·** .. ,n. Prove that for every x E (a, b) there exists ( such that .. , xn} < 5 < max{x, x0, x1, · f(a+1)(C) (n+ 1)! , Xn}, and f(x) – P(x) = (x – xo)(x – x1) ·. (x – xn). ... Hint: Fix x, and denote by M the number that satisfies the equation M ;(x – xo)(x – x1) · . (x – xn). (n + 1)! f(x) – P(x) = - | ... Furthermore, consider the function M g(t) = f(t) – P(t) - In i y(t – xo)(t – x1) · · · (t – xn), te (a, b), and observe that g(x) = g(x0) = g(xn) = 0. %3D =.. .- %3D
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