Let $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint xyz = 1. Then f has a relative minimum at

A) $( 1 , \pm 1 , \pm 1 )$
B) $( - 1 , \pm 1 , \mp 1 )$
C) $( 1,1,1 ) , ( - 1 , - 1 , - 1 )$
D) $( 1 , - 1 , - 1 ) , ( - 1,1 , - 1 )$
E) $( 1 , \pm 1 , \pm 1 ) , ( - 1 , \pm 1 , \mp 1 )$

Question

Let $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint xyz = 1. Then f has a relative minimum at

A) $( 1 , \pm 1 , \pm 1 )$
B) $( - 1 , \pm 1 , \mp 1 )$
C) $( 1,1,1 ) , ( - 1 , - 1 , - 1 )$
D) $( 1 , - 1 , - 1 ) , ( - 1,1 , - 1 )$
E) $( 1 , \pm 1 , \pm 1 ) , ( - 1 , \pm 1 , \mp 1 )$

Quizplus · Accepted Answer

Explanation:Using the method of Lagrange multipliers, we have$$
abla f = \lambda 
abla g$$$$2x = \lambda yz$$$$2y = \lambda xz$$$$2z = \lambda xy$$$$xyz = 1$$From the first three equations, we have$$x = y = z$$Substituting into the fourth equation, we get$$x^3 = 1$$$$x = y = z = 1$$or$$x = y = z = -1$$Therefore, the relative minimum occurs at$$( 1 , \pm 1 , \pm 1 ) , ( - 1 , \pm 1 , \mp 1 )$$

Let f(x,y,z)=x2+y2+z2f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }f(x,y,z)=x2+y2+z2

Let $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$