$\"\"$

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The product of two numbers is $\"\"$ .

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Let $\"\"$ and $\"\"$ be the two positive numbers.

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Therefore, $\"\"$ .

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The sum of first number and three times the second number is minimum,

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Consider above sum function as $\"\"$ .

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Thus, $\"\"$

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$\"\"$ is minimum.

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$\"\"$

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Substitute $\"\"$ value in $\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Differentiate the function $\"\"$

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$\"\"$

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Find the critical numbers by equating derivative to zero.

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ value in the expression $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ is minimumu when $\"\"$ and $\"\"$ .

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The two numbers are $\"\"$ and $\"\"$ .

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$\"\"$

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The two numbers are $\"\"$ and $\"\"$ .