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प्रश्न
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.
उत्तर
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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