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The sum of two positive real numbers x and y is 12. Find an x and y among these such that the product xy² is maximal.

X=
y =

User Efreeto
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2 Answers

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Step-by-step explanation:

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The sum of two positive real numbers x and y is 12. Find an x and y among these such-example-1
User Hung Tran
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Final answer:

To find the values of x and y that maximize the product xy², we need to maximize x and minimize y. The values of x and y that maximize the product xy² are x = 2 and y = 10.

Step-by-step explanation:

To find the values of x and y that maximize the product xy², we need to maximize x and minimize y.

Since the sum of x and y is 12, we can set up an equation:

x + y = 12

Solving for y, we get:

y = 12 - x

Substituting this expression for y in the product xy²:

xy² = x(12 - x)² = 12x - x³

To find the maximum value of xy², we need to find the critical points of this function. Taking the derivative and setting it equal to zero:

dy/dx = 12 - 3x² = 0

3x² = 12

x² = 4

x = ±2

Since x and y are positive real numbers, we can choose x = 2 and solve for y:

y = 12 - x = 12 - 2 = 10

Therefore, the values of x and y that maximize the product xy² are x = 2 and y = 10.

User Arun Gunalan
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