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5. Assume that f : (a, b) → R is an n + 1 times differentiable function. Fur- thermore, assume that f(xo) = f(x1) x0, X1, , Xn+1 € (a, b) such that x; x; for i # j. Prove that there exists ( such that min; x; < 5 < max; x;, and f(n+1)(S) = 0. = f(xn+1) = 0 for some points ... ... Hint: Successively apply the Mean Value Theorem to f and its derivatives.

5. Assume that f : (a, b) → R is an n + 1 times differentiable function. Fur- thermore, assume that f(xo) = f(x1) x0, X1, , Xn+1 € (a, b) such that x; x; for i # j. Prove that there exists ( such that min; x; < 5 < max; x;, and f(n+1)(S) = 0. = f(xn+1) = 0 for some points ... ... Hint: Successively apply the Mean Value Theorem to f and its derivatives.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Mean Value Theorem

5. Assume that f : (a, b) → R is an n + 1 times differentiable function. Fur- thermore, assume that f(xo) = f(x1) x0, X1, , Xn+1 € (a, b) such that x; x; for i # j. Prove that there exists ( such that min; x; < 5 < max; x;, and f(n+1)(S) = 0. = f(xn+1) = 0 for some points ... ... Hint: Successively apply the Mean Value Theorem to f and its derivatives.
Transcribed Image Text:5. Assume that f : (a, b) → R is an n + 1 times differentiable function. Fur- thermore, assume that f(xo) = f(x1) x0, X1, , Xn+1 € (a, b) such that x; x; for i # j. Prove that there exists ( such that min; x; < 5 < max; x;, and f(n+1)(S) = 0. = f(xn+1) = 0 for some points ... ... Hint: Successively apply the Mean Value Theorem to f and its derivatives.
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