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Suppose f is differentiable everywhere. Show that if æ* is a local minimizer for f, Vf(x*) = 0. Note that we are not assuming the gradient map is continuous.

Suppose f is differentiable everywhere. Show that if æ* is a local minimizer for f, Vf(x*) = 0. Note that we are not assuming the gradient map is continuous.

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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Exercise 1.2 ( : Let f: R" → R. (a) Suppose f is differentiable everywhere. Show that if a* is a local minimizer for f, Vf(x) = 0. Note that we are not assuming the gradient map is continuous. (b) Suppose f is twice continuously differentiable. By defining (n) formula (1) = (0) + - $¹(0) + ² * $"(t) dt da. Hence, show that for any a, s € R" f(x+ns), justify the \ƒ(x + s) − ƒ(x) − g(x)¹s − ½s¹H(x)s| ≤ 79||$||2, whenever f has a Lipschitz continuous Hessian with Lipschitz constant on S.
Transcribed Image Text:Exercise 1.2 ( : Let f: R" → R. (a) Suppose f is differentiable everywhere. Show that if a* is a local minimizer for f, Vf(x) = 0. Note that we are not assuming the gradient map is continuous. (b) Suppose f is twice continuously differentiable. By defining (n) formula (1) = (0) + - $¹(0) + ² * $"(t) dt da. Hence, show that for any a, s € R" f(x+ns), justify the \ƒ(x + s) − ƒ(x) − g(x)¹s − ½s¹H(x)s| ≤ 79||$||2, whenever f has a Lipschitz continuous Hessian with Lipschitz constant on S.
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