Express 18 as a sum of two positive numbers whose product of the first and square of the second is as large as possible.

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18 has to be expressed as the sum of two numbers x and y to maximize the value of x*y^2.

If x + y = 18

x = 18 - y

The product P = (18 - y)*y^2

=> 18y^2 - y^3

The extreme value of P is at `(dP)/(dy)...

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18 has to be expressed as the sum of two numbers x and y to maximize the value of x*y^2.

If x + y = 18

x = 18 - y

The product P = (18 - y)*y^2

=> 18y^2 - y^3

The extreme value of P is at `(dP)/(dy) = 0` .

`(dP)/(dy) = 36y - 3y^2`

36y - 3y^2 = 0

=> 3y(12 - y) = 0

=> y = 0 and y = 12

For y = 0 the product of x and y^2 is 0; this can be eliminated.

The required numbers that add up to 18 and where the product of the first number with the square of the other is maximum are 6 and 12.

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