$\"\"$

\

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

\

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

\

Therefore, $\"\"$ .

\

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

\

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

\

Thus, $\"\"$

\

$\"\"$ is minimum.

\

$\"\"$

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Differentiate the function $\"\"$ .

\

$\"\"$

\

Find the critical numbers by equating derivative to zero.

\

$\"\"$

\

$\"\"$ .

\

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

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

\

$\"\"$

\

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

\

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

\

$\"\"$

\

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