# Book is loading, please wait ....

# Step-by-Step Math Answer

Assign variables to the unknowns.

Let the two positive numbers be
*
x
*
and
*
y
*
.

Write a primary equation.

To find two positive numbers that satisfies the given conditions, first write a primary equation.

Here, the sum of the first number and three times the second number is to be minimized. So, write an equation for the sum.

The sum
*
S
*
is
*
x
*
+ 3
*
y
*
.

Therefore, a primary equation is

*
S
*
=
*
x
*
+ 3
*
y
*
.

Write a secondary equation.

To reduce the primary equation, we have to write a secondary equation.

So, write an equation for the product.

Here, the product
*
xy
*
is 192.

Therefore, a secondary equation is

*
xy
*
= 192.

Use the secondary equation to rewrite the primary equation.

Next, rewrite the primary equation in terms of a single variable, say
*
y
*
.

So, write the secondary equation in terms of
*
y
*
.

Substitute
for
*
x
*
in the primary equation.

*
*

Determine the feasible domain of the primary equation.

Now, determine the feasible domain of the primary equation.

So, find the values of
*
y
*
for which the problem makes sense.

Since the number
*
y
*
is positive and the product is 192, the feasible domain of the primary equation is

0
*
y
*
192.

Find the first derivative of the function
*
S
*
.

To find the critical number, first find the first derivative of the function
*
S
*
.

Set the derivative equal to 0 and solve for
*
y
*
.

Next, set the derivative equal to zero.

Now, solve for
*
y
*
.

*
y
*
= 8

So, the critical number is

*
y
*
= 8.

Make a table that shows the testing of two intervals determined by the critical number.

Make a table that shows the testing of two intervals determined by the critical number.

So,
*
f
*
is decreasing on the interval (0, 8) and increasing on the interval (8, 192).

A relative minimum occurs where
*
f
*
changes from decreasing to increasing.

A relative minimum occurs where
*
f
*
changes from decreasing to increasing.

From the table, the function changes from decreasing to increasing at the point where
*
y
*
= 8.

So, the function
*
S
*
is minimum at
*
y
*
= 8.

Substitute 8 for
*
y
*
in the secondary equation and simplify.

To find
*
x
*
, substitute 8 for
*
y
*
in the secondary equation.

*
x
*
· 8 = 192

Simplify.

*
x
*
= 24

So, the two positive numbers are 24 and 8.