Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. Let x denote the length of the side of the square being cut out. Let y denote the length of the base.
V = x · y² 2 x + y = 3 y = 3 - 2 x V = x · ( 3 - 2 x )² = x · ( 9 - 12 x + 4 x² ) = 9 x - 12 x² + 4 x³ V ` = 9 - 24 x + 12 x² = 3 ( 4 x² - 8 x + 3 ) = 3 ( 4 x² - 6 x - 2 x + 3 ) = = 3 [ 2 x ( 2 x - 3 ) - ( 2 x - 3 ) ] = 3 ( 2 x - 3 ) ( 2 x - 1 ) The largest volume is when: V ` = 0, so: 2 x - 3 = 0 2 x = 3 x = 1.5 ( which is incorrect, because : y = 0 ) or: 2 x - 1 = 0 2 x = 1 x = 0.5, y = 3 - 2 · 0.5 = 3 - 1 = 2 V max = 0.5 · 2² = 0.5 · 4 = 2 ft³
