Answered: 5. Set up the system of equations in… | bartleby

Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 17EQ

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5. Set up the system of equations in the Method of Lagrange Multipliers for the function f(x,
y, z) = xy + z and the constraint g(x, y, z) = x^4+y^4+ z^4=1. What must be true of a
point (x, y, z) that satisfies those equations?
Select one:
a. z=0
C b. x=y=0
c. X=y=z
C d. x=y or x=¯y
e. y=4x^3

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12.Consider the system dx/dt=x(a−σx−αy),dy/dt=y(−c+γx),dx/dt=xa−σx−αy,dy/dt=y−c+γx, where a, σ, α, c, and γ are positive constants. a.Find all critical points of the given system. How does their location change as σ increases from zero? Assume that a/σ > c/γ, that is, σ < aγ/c. Why is this assumption necessary? b.Determine the nature and stability characteristics of each critical point. c.Show that there is a value of σ between zero and aγ/c where the critical point in the interior of the first quadrant changes from a spiral point to a node. d.Describe the effect on the two populations as σ increases from zero to aγ/c.
Minimize ƒ(x, y, z) = xyz subject to the constraints x2 + y2 - 1 = 0 and x - z = 0

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