2 Answers

Best Answer

Step 1

It is given that the difference of two numbers is 100.

x-smaller

y-larger

xy=100

y=(100)/(x)

We write a function that represents the minimum sum of two numbers.

f(x,y)=x+y

Substitute y=100+x into f(x,y)=x+y.

f(x)=x+(100)/(x)

=(100x+100)/(x)

Now, we find the first derivative of the function f(x)=x+(100)/(x).

f'(x)=(x+(100)/(x))'

=(x)'+((100)/(x))'

=1-(100)/(x^(2))

Step 2

Find the critical points. Solve the equation f'(x)=0.

f'(x)=1-(100)/(x^(2))

0=1-(100)/(x^(2))

-1=-(100)/(x^(2))

x^(2)=100

x=sqrt(100)=10

x=-sqrt(100)=-10

Now, find the second derivative of the function f(x).

Use the second derivative test to determine whether the number we found was a critical number.

f""(x)=(1-(100)/(x^(2)))'

=(1)'-((100)/(x^(2)))'

=(200)/(x^(3))

Calculate f""(10).

f""(10)=(200)/(1000)=0.2>0

Since, it is positive, it means that yes, there is a minimum.

Calculate y. Substitute x=10 into xy=100.

10y=100

y=10

It is given that the difference of two numbers is 100.

x-smaller

y-larger

xy=100

y=(100)/(x)

We write a function that represents the minimum sum of two numbers.

f(x,y)=x+y

Substitute y=100+x into f(x,y)=x+y.

f(x)=x+(100)/(x)

=(100x+100)/(x)

Now, we find the first derivative of the function f(x)=x+(100)/(x).

f'(x)=(x+(100)/(x))'

=(x)'+((100)/(x))'

=1-(100)/(x^(2))

Step 2

Find the critical points. Solve the equation f'(x)=0.

f'(x)=1-(100)/(x^(2))

0=1-(100)/(x^(2))

-1=-(100)/(x^(2))

x^(2)=100

x=sqrt(100)=10

x=-sqrt(100)=-10

Now, find the second derivative of the function f(x).

Use the second derivative test to determine whether the number we found was a critical number.

f""(x)=(1-(100)/(x^(2)))'

=(1)'-((100)/(x^(2)))'

=(200)/(x^(3))

Calculate f""(10).

f""(10)=(200)/(1000)=0.2>0

Since, it is positive, it means that yes, there is a minimum.

Calculate y. Substitute x=10 into xy=100.

10y=100

y=10

Best Answer

Let the numbers be x,y
xy=100, y=(100)/(x)
f(x,y)=x+y
f(x)=x+(100)/(x)
Minimum of f(x) is obtained at f'(x)=0
-> f'(x)=1-(100)/(x^(2))=0
x=pm 10
therefore x=10, y=10