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Question
Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
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Solution
Let two numbers be x and y then
x + y = 5 ...(i)
Let S = x3 + y3 ...(ii)
= x3 + (5 – x)3 ...[From (i)]
`(dS)/dx` = 3x2 + 3(5 – x)2 (– 1)
`(dS)/dx` = 3x2 – 3(25 + x2 – 10x)
= 3x2 – 75 – 3x2 + 30x
= 30x – 75
For maximum or minimum
`(dS)/dx` = 0
`\implies` 30x – 75 = 0
`\implies` x = `75/35 = 5/2`
When x = `5/2`, y = `5 - 5/2 = 5/2` ...[From (i)]
`(d^2S)/(dx^2)` = 30 which is +ve.
So the sum is least when x = `5/2` and y = `5/2`
S = x2 + y2
= `25/4 + 25/4`
= `50/4`
= `25/2`
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