Let $f ( x , y , z ) = x + y + z$ with constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Then f has a relative maximum at

A) $( \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$
B) $( \sqrt { 3 } , - \sqrt { 3 } , \sqrt { 3 } )$
C) $( \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$
D) $( - \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$
E) $( - \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$

Question

Let $f ( x , y , z ) = x + y + z$ with constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Then f has a relative maximum at

A) $( \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$
B) $( \sqrt { 3 } , - \sqrt { 3 } , \sqrt { 3 } )$
C) $( \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$
D) $( - \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$
E) $( - \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$

Quizplus · Accepted Answer

The function $f(x, y, z) = x + y + z$ will have a relative maximum when $x, y,$ and $z$ are all positive and as large as possible under the given constraint $x^2 + y^2 + z^2 = 9$. The point $(\sqrt{3}, \sqrt{3}, \sqrt{3})$ satisfies this condition, maximizing $f$ while adhering to the constraint.

Let f(x,y,z)=x+y+zf ( x , y , z ) = x + y + zf(x,y,z)=x+y+z

Let $f ( x , y , z ) = x + y + z$