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If the product of two numbers is 25, is it possible to determine a minimum value for...

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xataexa | Student, Kindergarten | Salutatorian

Posted June 2, 2012 at 6:52 AM via web

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If the product of two numbers is 25, is it possible to determine a minimum value for the sum of the numbers?

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted June 2, 2012 at 6:59 AM (Answer #1)

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The product of two numbers is given to be 25. If one of the numbers is x, the other is `25/x` . The sum of the two numbers is `S = x + 25/x`

Now, a minimum value of x can be found at S' = 0 and if S'' is positive.

`S' = 1 - 25/x^2`

`S'' = 25/x^3`

Solving S' = 0

=> `25/x^2 = 1`

=> x = +5 and x = -5

For x = -5, S'' is negative and for x = 5, S'' is positive.

The minimum value for the sum of the numbers is at x = 5 and is equal to 10.

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dylee | High School Teacher | eNoter

Posted June 4, 2012 at 11:57 AM (Answer #2)

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Let two number be x and y

x*y=25
=> y = 25/x

x+y = x + 25/x

We can use the arithmetic-geometric mean formula, which states that 
(a+b)/2>=(ab)^(1/2)
=> a+b>=2*(ab)^(1/2)

plugging in x and 25/x for a and b,

x+25/x >=2*(x*25/x)^(1/2) = 2 * 25^(1/2) = 10

Therefore, the minimum value of the sum of two number x and y is 10.

And the minimum is achieved when x = y 
=> x = 25/x 
=> x^2 = 25
=> x = 5

x+y = 10 when x and y are both 5

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