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Problem 1. Prove part (2) of Theorem 6.14 of the course notes, which is given below. Theorem (Second Derivative Test). Suppose that f: R" → R is C³ in a neighbourhood of a critical point a ЄR". Let A₁₂ <

Problem 1. Prove part (2) of Theorem 6.14 of the course notes, which is given below. Theorem (Second Derivative Test). Suppose that f: R" → R is C³ in a neighbourhood of a critical point a ЄR". Let A₁₂ <

Calculus: Early Transcendentals
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 part (2) of Theorem 6.14 of the course notes, which is given below. Theorem (Second Derivative Test). Suppose that f: R" → R is C³ in a neighbourhood of a critical point a ЄR". Let A₁₂ <<n 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 part (2) of Theorem 6.14 of the course notes, which is given below. Theorem (Second Derivative Test). Suppose that f: R" → R is C³ in a neighbourhood of a critical point a ЄR". Let A₁₂ <<n 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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