Question
Find two positive numbers a and y such that their sum is 35 and the product x2y5 is a maximum.
Solution
Here x + y = 35 ⇒ y = 35 – x ...(1)
Let f (x) = x2 y5 = x2(35 – x)5
∴ f ' (x) = a2· 5 (35 – x)4 (– 1) + (35 – x)5 · 2x
= x (35 – x)4 [ – 5x + 2 (35 – x)] = x (35 – x)4 (– 7x + 70)
= – 7x (35 – x)4 (x – 10)
f ' (x) = 0 ⇒ – 7x (35 – x)4 (x – 10)
⇒ x (x – 10) (35 – x)4 = 0 ⇒ x = 0, 10, 35
Rejecting x = 0. 35 as 0 < x < 35, we get, x = 10
When x < 10 slightly, f ' (x) = – (+)(+) (–) = + ve
When x > 10 slightly , f ' (x) = – (+) (+) (+) = – ve
∴ at x = 10, f ' (x) changes from +ve to negative
∴ f (x) has maximum value at a = 10, y = 35 – 10 = 25
∴ x = 10, y = 25
