Find the critical points of the function $f ( x , y , z ) = ( x - 4 ) ^ { 2 } + ( 7 - y ) ^ { 2 } + ( z - 2 ) ^ { 2 }$ , and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.
A)relative minimum at $( - 4 , - 7 , - 2 )$
B)relative maximum at $( - 4 , - 7 , - 2 )$
C)relative minimum at $( 4,7,2 )$
D)relative maximum at $( 4,7,2 )$
E)no relative extrema

Question

Quizplus · Accepted Answer

The function $f(x, y, z) = (x - 4)^2 + (7 - y)^2 + (z - 2)^2$ is a sum of squares, which is minimized when each square is zero. This occurs at $(x, y, z) = (4, 7, 2)$. Since the function is a sum of squares, it can only increase from this point, indicating a relative minimum.

Find the Critical Points of the Function f(x,y,z)=(x−4)2+(7−y)2+(z−2)2f ( x , y , z ) = ( x - 4 ) ^ { 2 } + ( 7 - y ) ^ { 2 } + ( z - 2 ) ^ { 2 }f(x,y,z)=(x−4)2+(7−y)2+(z−2)2

Find the Critical Points of the Function $f ( x , y , z ) = ( x - 4 ) ^ { 2 } + ( 7 - y ) ^ { 2 } + ( z - 2 ) ^ { 2 }$