Step 1:

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The three positive integers is $\"\"$ and $\"\"$ .

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The sum of three positive integers is $\"\"$ .

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The sum of squares is $\"\"$ .

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ .

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Apply partial derivative on each side with respect to $\"\"$ .

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$\"\"$ .

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$\"\"$

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$\"\"$

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The sum of squares is minimum so, set $\"\"$ .

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$\"\"$

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$\"\"$ .

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Step 2: \ \

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$\"\"$

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Apply partial derivative on each side with respect to $\"\"$ .

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$\"\"$ .

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$\"\"$

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$\"\"$

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The sum of squares is minimum so, set $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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The three positive integers are $\"\"$ , $\"\"$ and $\"\"$ .

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Solution:

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The three positive integers are $\"\"$ , $\"\"$ and $\"\"$ . \ \