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