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 For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter9: Multivariable Calculus

Section9.CR: Chapter 9 Review

Problem 4CR

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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.

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