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Let n >1. Suppose that f has (n – 1)-th order continuous derivative and is n-times differentiable in IR, and that f(x) = 0 has (n + 1) distinct roots x1 < x2 <……< xn+1· Show that there exists c E (a, b) such that f(n) (c) = 0. Hint: apply Rolle's Theorem for n times; or use induction on n to have a better proof structure.

Let n >1. Suppose that f has (n – 1)-th order continuous derivative and is n-times differentiable in IR, and that f(x) = 0 has (n + 1) distinct roots x1 < x2 <……< xn+1· Show that there exists c E (a, b) such that f(n) (c) = 0. Hint: apply Rolle's Theorem for n times; or use induction on n to have a better proof structure.

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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Let n > 1. Suppose that f has (n – 1)-th order continuous derivative and is n-times differentiable in R, and that f(x) = 0 has (n + 1) distinct roots x1 < x2 <……< xn+1· Show that there exists c E (a, b) such that f(n) (c) = 0. Hint: apply Rolle's Theorem for n times; or use induction on n to have a better proof structure.
Transcribed Image Text:Let n > 1. Suppose that f has (n – 1)-th order continuous derivative and is n-times differentiable in R, and that f(x) = 0 has (n + 1) distinct roots x1 < x2 <……< xn+1· Show that there exists c E (a, b) such that f(n) (c) = 0. Hint: apply Rolle's Theorem for n times; or use induction on n to have a better proof structure.
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