A Closed Rectangular Box with Square Ends Is to Be $x ^ { 2 }$

Multiple Choice

A closed rectangular box with square ends is to be designed so that the surface area of the box is minimized. [Note: Surface area = 2 $x ^ { 2 }$ + 4xy.] It is required that the volume be 32 cubic inches. Which of the following is the Lagrange function F(x, y, λ) for this problem?

A) 2 $x ^ { 2 }$ + 4xy + λ(32)
B) ( $x ^ { 2 }$ y - 32) + λ(2 $x ^ { 2 }$ + 4xy)
C) 2 $x ^ { 2 }$ + 4xy + λ( $x ^ { 2 }$ y - 32)
D) $x ^ { 2 }$ y + λ(2 $x ^ { 2 }$ + 4xy - 32)
E) none of these