If the product of 2 numbers is 48, minimum positive value of the sum?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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Let the two numbers be x and y.

We have their product x*y equal to 48.

xy = 48

=> x = 48/y

The sum S = x + y can be written as S = y + 48/y.

Now let us differentiate S with respect to y.

S’ = 1 – 48/y^2

Equating this to 0

1 – 48/y^2 = 0

=> 48/y^2 = 1

=> y^2 = 48

=> y = 4 sqrt 3 and y = -4 sqrt 3

At y = 4 sqrt 3, x= 12/sqrt 3, the sum x + y = 4 sqrt 3 + 12/ sqrt 3

At y = -4qrt 3, x = -12/ sqrt 3, the sum x + y = - 4sqrt 3 – 12/ sqrt 3

Also S’’ = 96/y^3, at, and is positive for y = +4 sqrt 3. So we have a point of minimum at y = 4 sqrt 3 in terms of positive values of the sum.

Therefore the minimum positive value of the sum is 4 sqrt 3 + 12/ sqrt 3.

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