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# Product of two numbers?

a) Find the minimum product of two numbers whose difference is 17.

b) What are the two numbers?

Update:

I need help in solving this, besides trial and error.

### 9 Answers

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• You have two numbers that has a difference of 17:

x - y = 17

If we solve this for y in terms of x:

-y = 17 - x

y = x - 17

Then you want to find the minimum product of the two numbers:

f(x, y) = xy

If we substitute the expression into the function we have a function with only one unknown:

f(x) = x(x - 17)

f(x) = x² - 17x

The x that makes f(x) a minimum can be found by solving for the zero of the first derivative:

f'(x) = 2x - 17

0 = 2x - 17

-2x = -17

x = 17/2 or 8.5

Now we can solve for y:

y = x - 17

y = 8.5 - 17

y = -8.5

The two numbers are:

-8.5 and 8.5

And that minimum product is: -72.25

• If you are asked to work out the product of two or more numbers, then you need to multiply the numbers together. If you are asked to find the sum of two or more numbers, then you need to add the numbers together. Below, we will work through several examples together. "Product" means multiply.

• The question wants the minimum product of the two numbers. That is 18.

If the numbers are 18 and 1 the difference is seventeen, which meets the second of the question requirements.

All the rest is BS.

You should always consider the special case of ,1, and ,#, before diving into a sea of BS.

• a - b = 17

a = b + 17

p = ab

p = (b + 17).b

p = b² + 17b ← this is a function of b → minimum when the derivative is zero

p' = 2b + 17 → then you solve for b the equation: p' = 0

2b + 17 = 0

b = - 17/2

Recall: a = b + 17

a = - (17/2) + 17

a = 17/2

p = ab

p = - 289/4

• Let x & y be the 2 numbers, x>y then

x-y=17=>x=y+17

P=xy

=>

P=(y+17)y

P=y^2+17y

P'=2y+17

P"=2>0

P'=0

=>

2y+17=0

=>

y=-17/2=-8.5

x=17+(-17/2)=17/2=8.5

=>

min. P=-72.25 for the finite x,y.

• Let the mean of two numbers be x. So two numbers are x ± 17/2 and their product becomes x^2 - 289/4.

If "numbers" are "real numbers" then we set x = 0, two numbers are -17/2 and 17/2, their product is -289/4 = -72.25.

If "numbers" are "integers" then we set x = -1/2 or 1/2, two numbers are -9 and 8 or -8 and 9, their product is -72.

If "numbers" are other numbers, what are they ? For example, positive integers ?

• The two numbers are 1 and 18.

• Hint: Apply the same reason he's already given to you in your other similar posts.

Show your work in progress here. Then we'll take it from there.

• 18

The numbers are 1 and 18

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