1. Let f: (a, b) → R be a smooth function (i.e. derivatives of all orders exist). Suppose that (a, b) is a point where the first 2k-1 derivatives vanish, xo f'(x) = f'(xo) = f(2k-¹) (0) = 0, while f(2k) (x) > 0. Here 2k is a positive even integer. Prove that f(ro) is a local minimum of f, i.e. for some r> 0 we have f(ro) ≤ f(x) for all xo - r < x < xo + r.
1. Let f: (a, b) → R be a smooth function (i.e. derivatives of all orders exist). Suppose that (a, b) is a point where the first 2k-1 derivatives vanish, xo f'(x) = f'(xo) = f(2k-¹) (0) = 0, while f(2k) (x) > 0. Here 2k is a positive even integer. Prove that f(ro) is a local minimum of f, i.e. for some r> 0 we have f(ro) ≤ f(x) for all xo - r < x < xo + r.
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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Transcribed Image Text:1. Let f: (a, b) → R be a smooth function (i.e. derivatives of all orders exist). Suppose that
(a, b) is a point where the first 2k - 1 derivatives vanish,
xo
f'(x) = f'(x) = = f(2k-¹) (0) :
==
= 0,
while f(2k) (x) > 0. Here 2k is a positive even integer. Prove that f(ro) is a local minimum of
f, i.e. for some r> 0 we have f(xo) ≤ f(x) for all xo − r < x < xo + r.
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