Multiple Choice Questions
Short Answer TypeDetermine two positive numbers whose sum is 15 and the sum of whose, squares is minimum.
Amongst all pairs of positive numbers with product 64, find those whose sum is the least.
Find two positive numbers a and y such that their sum is 35 and the product x2y5 is a maximum.
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
Find two positive numbers a and y such that their sum is 35 and the product x2y5 is a maximum.
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