f(x, y, z, w) = a cosz+acosy+acosz+acos w+zy+yz+zw which is a function of four variables, and it has a critical point at x=y=z=w = 0 (this is obvious; you do not need to verify it). (A) Classify the critical point, assuming a = 0. (B) Classify the critical point, assuming a = 1/2. (C) Classify the critical point, assuming a = 1.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 93E
Question
This problem is about the function
f(x, y, z, w) = a cos x + a cosy + a cosz+acos w+ xy + yz+zw
which is a function of four variables, and it has a critical point at
x=y=z=w=0 (this is obvious; you do not need to verify it).
(A) Classify the critical point, assuming a = 0.
(B) Classify the critical point, assuming a = 1/2.
(C) Classify the critical point, assuming a = 1.
(D) Classify the critical point, assuming a = √2.
(E) Classify the critical point, assuming a = 2.
(F) Determine the values of a for which the second derivative test fails, at the
given critcal point.
Transcribed Image Text:This problem is about the function f(x, y, z, w) = a cos x + a cosy + a cosz+acos w+ xy + yz+zw which is a function of four variables, and it has a critical point at x=y=z=w=0 (this is obvious; you do not need to verify it). (A) Classify the critical point, assuming a = 0. (B) Classify the critical point, assuming a = 1/2. (C) Classify the critical point, assuming a = 1. (D) Classify the critical point, assuming a = √2. (E) Classify the critical point, assuming a = 2. (F) Determine the values of a for which the second derivative test fails, at the given critcal point.
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