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# 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 product is to be maximized. So, write an equation for the product.

The product
*
P
*
is
*
xy
*
.

Therefore, a primary equation is

*
P
*
=
*
xy
*
.

Write a secondary equation.

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

So, write an equation for the sum.

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

Therefore, a secondary equation is

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

Use the secondary equation to rewrite the primary equation.

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

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

*
y
*
=
*
S
*
–
*
x
*
.

Substitute
*
S
*
–
*
x
*
for
*
y
*
in the primary equation and simplify.

*
P
*
=
*
x
*
(
*
S
*
–
*
x
*
) or
*
Sx
*
–
*
x
*
^{
2
}
.

Determine the feasible domain of the primary equation.

Now, determine the feasible domain of the primary equation.

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

Since the number
*
x
*
is positive, the feasible domain of the primary equation is

*
x
*
0.

Find the first derivative of the function
*
P
*
.

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

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

Next, set the derivative equal to 0.

*
S
*
– 2
*
x
*
= 0

Now, solve for
*
x
*
.

So, the critical number is .

Find the second derivative of the function
*
P
*
.

Find the second derivative of the function
*
P
*
.

Use the Second Derivative Test.

Here, when .

So, by the Second Derivative Test, the function
*
P
*
is maximum at
.

Substitute
for
*
x
*
in the secondary equation and simplify.

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

Simplify.

So, the two positive numbers are .