Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$ with constraint xy = 1. Then f has a relative minimum at

A) $\left( \sqrt { 2 } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$
B) $\left( - \sqrt { 2 } , - \frac { 1 } { \sqrt { 2 } } , 0 \right)$
C) $\left( \frac { 1 } { \sqrt { 2 } } , \sqrt { 2 } , 0 \right)$
D) $\left( \pm \sqrt { 2 } , \pm \frac { 1 } { \sqrt { 2 } } , 0 \right)$
E) $\left( - \frac { 1 } { \sqrt { 2 } } , - \sqrt { 2 } , 0 \right)$

Question

Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$ with constraint xy = 1. Then f has a relative minimum at

A) $\left( \sqrt { 2 } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$
B) $\left( - \sqrt { 2 } , - \frac { 1 } { \sqrt { 2 } } , 0 \right)$
C) $\left( \frac { 1 } { \sqrt { 2 } } , \sqrt { 2 } , 0 \right)$
D) $\left( \pm \sqrt { 2 } , \pm \frac { 1 } { \sqrt { 2 } } , 0 \right)$
E) $\left( - \frac { 1 } { \sqrt { 2 } } , - \sqrt { 2 } , 0 \right)$

Quizplus · Accepted Answer

The function $f(x, y, z)$ is minimized when $z = 0$ and $xy = 1$. The points $(\sqrt{2}, \frac{1}{\sqrt{2}}, 0)$ and $(- \sqrt{2}, -\frac{1}{\sqrt{2}}, 0)$ satisfy these conditions, giving the relative minimum.

Let f(x,y,z)=x2+4y2+16z2f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }f(x,y,z)=x2+4y2+16z2

Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$