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Problem 1. Prove Theorem 6.14 which is given below. Theorem (Second Derivative Test). Suppose that ƒ : R³ → R is C³ in a neighbourhood of a critical point a = Rn. Let λ1 ≤ №2 < ... < An be the eigenvalues of D²f(a). Then: (2) If all the eigenvalues are negative, then a is a strict local maximum of f. Hint: You should be able to draw inspiration from the proof of (1), which is given in the courseware.

Problem 1. Prove Theorem 6.14 which is given below. Theorem (Second Derivative Test). Suppose that ƒ : R³ → R is C³ in a neighbourhood of a critical point a = Rn. Let λ1 ≤ №2 < ... < An be the eigenvalues of D²f(a). Then: (2) If all the eigenvalues are negative, then a is a strict local maximum of f. Hint: You should be able to draw inspiration from the proof of (1), which is given in the courseware.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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Problem 1. Prove Theorem 6.14 which is given below. Theorem (Second Derivative Test). Suppose that ƒ : R³ → R is C³ in a neighbourhood of a critical point a = Rn. Let λ1 ≤ №2 < ... < An be the eigenvalues of D²f(a). Then: (2) If all the eigenvalues are negative, then a is a strict local maximum of f. Hint: You should be able to draw inspiration from the proof of (1), which is given in the courseware.
Transcribed Image Text:Problem 1. Prove Theorem 6.14 which is given below. Theorem (Second Derivative Test). Suppose that ƒ : R³ → R is C³ in a neighbourhood of a critical point a = Rn. Let λ1 ≤ №2 < ... < An be the eigenvalues of D²f(a). Then: (2) If all the eigenvalues are negative, then a is a strict local maximum of f. Hint: You should be able to draw inspiration from the proof of (1), which is given in the courseware.
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