Question

Find the positive integers x and y such that x + y = 60 and xy3 is maximum.

Answer 1

Given x + y = 60

⇒ y = 60 – x

Let * = xy3

= x (60 – x)3

ds/dx = x . 3(60 – x)2 (0 – 1) + (60 – x)3 . 1

= –3x(60 – x)2 + (60 – x)3

= (60 – x)2 + (–3x + 60 – x)

= (60 – x)2 (60 – 4x)

ds/dx = 0 ⇒ (60 – x)2 (60 – 4x) = 0

⇒ 60 – 4x = 0

⇒ 4x = 60

⇒ x = 15

d2s/dx2 = (60 – x)2 (–4) + (60 – 4x) 2 (60 – x) (–1)

= – 4 (60 –x)2 – 2 (60 – 4x) (60 – x)

∴ * is maximum at x = 15

∴ y = 60 – 15 = 45

Hence x = 15, y = 45.