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Let xo,..., xn be n + 1 distinct points. Consider a function f(x) and assume there exists a polynomial p(x) of degree at most n + 1 such that p(®k) = f(xk) for k = 0,., n, and p'(xn) = f'(xn). Show that this polynomial is unique.

Let xo,..., xn be n + 1 distinct points. Consider a function f(x) and assume there exists a polynomial p(x) of degree at most n + 1 such that p(®k) = f(xk) for k = 0,., n, and p'(xn) = f'(xn). Show that this polynomial is unique.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. а) Let xo, . . . , xn be n + 1 distinct points. Consider a function f(x) and assume there exists a polynomial p(x) of degree at most n + 1 such that p(xk) = f(xk) for k = 0,..., n, and p'(xn) = f'(xn). Show that this polynomial is unique. b) satisfies p(k) = 0 for k = 0,.., n, and p'(n) = 1. Use divided differences to find the polynomial p(x) of degree n+1 which || || .. .
Transcribed Image Text:3. а) Let xo, . . . , xn be n + 1 distinct points. Consider a function f(x) and assume there exists a polynomial p(x) of degree at most n + 1 such that p(xk) = f(xk) for k = 0,..., n, and p'(xn) = f'(xn). Show that this polynomial is unique. b) satisfies p(k) = 0 for k = 0,.., n, and p'(n) = 1. Use divided differences to find the polynomial p(x) of degree n+1 which || || .. .
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