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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.

It is given that the sum of the first number and twice the second number is 100.

So, a secondary equation is

*
x
*
+ 2
*
y
*
= 100.

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
*
.

*
x
*
= 100 – 2
*
y
*

Substitute 100 – 2
*
y
*
for
*
x
*
in the primary equation and simplify.

*
P
*
= (100 – 2
*
y
*
)
*
y
*
or 100
*
y
*
– 2
*
y
*
^{
2
}

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 can be at most 50, the feasible domain of the primary equation is

0
*
y
*
50.

Find the derivative of the function
*
P
*
.

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

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

Next, set the derivative equal to zero.

100 – 4
*
y
*
= 0

Now, solve for
*
y
*
.

*
y
*
= 25

So, the critical number is

*
y
*
= 25.

Find the second derivative of the function
*
P
*
.

Next, find the second derivative of the function
*
P
*
.

Use the Second Derivative Test.

Here,
when
*
y
*
= 25.

So, by the Second Derivative Test, the function
*
P
*
is maximum at
*
y
*
= 25.

Substitute 25 for
*
y
*
in the secondary equation and simplify.

To find
*
x
*
, substitute 25 for
*
y
*
in the secondary equation.

*
x
*
+ 2(25) = 100

Simplify.

*
x
*
= 50

So, the two positive numbers are 50 and 25.