Answered: f(y) - f(z) = [²√ 1 (² + 1 dt = So…

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f(y) - f(z) = [²√ 1 (² + 1 dt = So f(z+t(y-2)) dt = ₁ Df (z+t(y-2)) dt (y-2).

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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Question

Let f be continuously differentiable: ℝⁿ → ℝⁿ
.
Prove that (attached image)

and prove that ||f(y)-f(z)|| ≤ C ||y-z|| if || D f(u) ||≤ C for all u on the line segment between y
and z.

please if able write some explanation with the taken steps, thank you in advance.

d
= √² = f( ² +
dt
f(y) - f(²)=√√
= [² D
f(z+t(y-2)) dt =
Df (z+t(y-2)) dt (y - z).

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