Let k(x) = h(x) - g(x), where g and h are infinitely differentiable functions from R to R. Define f(n) to be the nth derivative of a function mapping R to R. Suppose hn(x) = gn(x) for all x. Suppose k(x1) = k(x2) = ... = k(xn) = 0 forx1 < x2 < x3 < ... < xn . Show that g = h.

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Description: Let k(x) = h(x) - g(x), where g and h are infinitely differentiable functions from R to R. Define f(n) to be the nth derivative of a function mapping R to R. Suppose hn(x) = gn(x) for all x. Suppose k(x1) = k(x2) = ... = k(xn) = 0 forx1 < x2 < x3 <

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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 k(x) = h(x) - g(x), where g and h are infinitely differentiable functions from R to R. Define f⁽ⁿ⁾ to be the n^th derivative of a function mapping R to R. Suppose hⁿ(x) = gⁿ(x) for all x. Suppose k(x₁) = k(x₂) = ... = k(x_n) = 0 for

x₁ < x₂< x₃ < ... < x_n. Show that g = h.

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Let $k (x) = h (x) - g (x)$ , where $g$ and $h$ are infinitely differentiable functions from $ℝ \to ℝ$ .

Let $f^{(n)}$ be the $n^{t h}$ derivative of a function mapping $ℝ \to ℝ$ and $h^{(n)} (x) = g^{(n)} (x), \forall x \in ℝ$ .

Suppose $k (x_{1}) = k (x_{2}) = \cdot \cdot \cdot = k (x_{n}) = 0$ for $x_{1} < x_{2} < \cdot \cdot \cdot < x_{n}$

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