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Suppose at x = x, Vf(x) = 0. What can you say about x if (a) f(x) is convex? (b) f(x) is concave? (c) V²f(x) is indefinite? (d) V²f(x) is positive semidefinite? (e) V²f(x) is negative semidefinite?

Suppose at x = x, Vf(x) = 0. What can you say about x if (a) f(x) is convex? (b) f(x) is concave? (c) V²f(x) is indefinite? (d) V²f(x) is positive semidefinite? (e) V²f(x) is negative semidefinite?

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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**3.14.** Suppose at \( x = \bar{x}, \nabla f(\bar{x}) = 0 \). What can you say about \( \bar{x} \) if (a) \( f(x) \) is convex? (b) \( f(x) \) is concave? (c) \( \nabla^2 f(\bar{x}) \) is indefinite? (d) \( \nabla^2 f(\bar{x}) \) is positive semidefinite? (e) \( \nabla^2 f(\bar{x}) \) is negative semidefinite?
Transcribed Image Text:**3.14.** Suppose at \( x = \bar{x}, \nabla f(\bar{x}) = 0 \). What can you say about \( \bar{x} \) if (a) \( f(x) \) is convex? (b) \( f(x) \) is concave? (c) \( \nabla^2 f(\bar{x}) \) is indefinite? (d) \( \nabla^2 f(\bar{x}) \) is positive semidefinite? (e) \( \nabla^2 f(\bar{x}) \) is negative semidefinite?
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