0

A company wants to encrypt a document containing important passwords. To do this, the product of a positive integer and a negative integer will need to be minimized. If the positive integer is 11 greater than the negative integer, what is the minimum product of the two numbers?

I really don't know where to start here, but if p= positive integer, n=negative integer, does p=11n? If this is right, where do I go from here?

Ayanoria
  • 89
  • There are very few cases to try! How many examples can you find of a positive and a negative integer that differ by $11$? – lulu Aug 18 '20 at 23:25

1 Answers1

1

A company wants to encrypt a document containing important passwords. To do this, the product of a positive integer and a negative integer will need to be minimized.

We can't do algebra (or arithmetic) without symbols. We have just been told that we have two quantities. We were not given symbolic names for them, so we are responsible for assigning them names.

"Let $p$ be the positive integer and $n$ be the negative integer."

The particular names are not important. I used the names you used. If the symbols are easy to associate with the quantity in the setting which they represent, they can be easier for you and a reader to use. "$p$" for "positive" and "$n$" for "negative" are easily associated with the contextual quantities.

If the positive integer is 11 greater than the negative integer,

"We are told $p = 11+ n$."

what is the minimum product of the two numbers?

"We wish to minimize $pn$."

So the problem is to minimize $pn$ subject to $p = 11 + n$.

I assume you know how to do this? If not, post a comment and I'll continue.

Eric Towers
  • 67,037
  • oh I don't know how to do this, I'm not very good at optimization – Ayanoria Aug 18 '20 at 23:10
  • @Ayanoria : As given, there are two variables. Can we easily turn the objective function (the function that we want to maximize or minimize) into a single-variable function, by using the constraint? – Eric Towers Aug 18 '20 at 23:11
  • yes, so if $p=11+n$, the $pn$ would turn into $11n+n^2$. But what do we do about n being a negative integer and how do we optimize this? – Ayanoria Aug 18 '20 at 23:16
  • 1
    @Ayanoria : Do you recognize that the graph of this function of $n$ is a parabola? Does this parabola open up or down? Do we want to keep the entire parabola, or are we only interested in part of the parabola? – Eric Towers Aug 18 '20 at 23:31