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

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

Question

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

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

Quizplus · Accepted Answer

To find the relative minimum of $f(x, y, z) = x^2 + 4y^2 + 16z^2$ subject to the constraint $xyz = 1$, we can use the method of Lagrange multipliers. The constraint $xyz = 1$ does not specify the signs of $x$, $y$, and $z$, allowing them to be positive or negative as long as their product is positive. The symmetry in the options suggests that for any given solution $(x, y, z)$, flipping the signs of any two variables (or keeping them the same) will also be a solution, as long as the product $xyz = 1$ is maintained. This is because the function $f(x, y, z)$ only depends on the squares of $x$, $y$, and $z$, making it even in each variable and insensitive to their signs. Therefore, option E correctly lists all combinations of signs for the variables $x$, $y$, and $z$ in the given solutions, reflecting the symmetry and the constraint.

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