Find a positive number such that the sum of the number and its reciprocal is as small as possible.

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- Apr 8th 2007, 06:29 PMcamherokidOptimization Problem.
Find a positive number such that the sum of the number and its reciprocal is as small as possible.

- Apr 8th 2007, 06:38 PMJhevon
Let the positive number be x. So we have x > 0

we want x + 1/x to be a minimum.

let y = x + 1/x

=> y' = 1 - 1/x^2

for min, set y' = 0

=> 1 - 1/x^2 = 0

=> 1/x^2 = 1

=> x^2 = 1

=> x = +/- 1

=> x = 1, since x > 0

you can check +1 and -1 using the second derivative test if you want to be sure of this, or graph x + 1/x