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.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe60e14a8-856f-447c-ac90-795ae43e00b4%2F11e6599d-d7b6-46ce-a7c5-c3711e958ab7%2Fnd07k98_processed.png&w=3840&q=75)
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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