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Question
Find two positive numbers whose product is 20 and their sum is minimum
Solution
Let the two positive numbers be ‘x’ and ‘y’
Given product is 20
⇒ xy = 20
⇒ y = `20/x`
Sum S = x + y
S = `x + 20/x`
`"dS"/("d"x) = 1 - 20/x^2`
For maximum or minimum, `"dS"/("d"x)` = 0
x2 – 20 = 0
x2 = 20
x = `+ 2sqrt(5)`
x = `- 2sqrt(5)` is not possible
`("d"^2"S")/("d"x^2) = 40/x^3`
At x = `2sqrt(5), ("d"^2"S")/("d"x^2) > 0`
∴ Sum ‘S’ is minimum when x = `2sqrt(5)`
y = `20/(2sqrt(5)) = 2sqrt(5)`
Minimum sum = `2sqrt(5) + 2sqrt(5)`
= `4sqrt(5)`
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