Question

The sum of two positive numbers is 10. Find the numbers if their squares is a minimum

Answer 1

Here's one way to do it, but it's not what your looking for.

Let x and y be a positive even integer. Then x = 2k for some integer k, y= 2L for some integer L.

x + y = 10
(2k) + (2L) = 10
2(k+L) = 10

exp:
2(3+2) = 10

Answer 2

Thanks StarJ. Although I don't think this is what I'm looking for. currently doing optimisation in year 12 maths B. I know x + y = 10, therefore y = 10 - x but not sure where to go from there.. thanks anyway :)

Answer 3

It's not clear to me what "...if their squares is a minimum" means.

Answer 4

@siklad96 - you have started correctly by stating that:\[x+y=10\]\[\therefore y=10-x\]Now the question states that you must minimize the sum of their squares.
So lets call the sum of the squares some function \(f\) given by:\[f=x^2+y^2\]The aim is to minimize \(f\) - understand so far?

Answer 5

yes, I understand that because there are two squares therefore x and y are squared, please continue :)

Answer 6

good. Now the first step is to reduce the number of variables. From the first condition given to you, we know that:\[y=10-x\]We can therefore substitute this into the expression for \(f\) to obtain:\[f=x^2+y^2=x^2+(10-x)^2=x^2+100-20x+x^2=2x^2-20x+100\]
Now, do you know how to find the minimum value for \(f\)?

Answer 7

I'm pretty sure its the quadratic formula? or you can factorise? correct me if I'm wrong, and how did you end up with \[x^2 + 100 - 20x\]?

Answer 8

\[(10-x)^2=(10-x)(10-x)=100-10x-10x+x^2=100-20x+x^2\]

Answer 9

so we end up with a quadratic equation (as you correctly recognised):\[f=2x^2-20x+100\]Do you know how to find the minimum value for an equation like this?

Answer 10

What optimisation techniques have you been taught?

Answer 11

\[-b \sqrt{b^2 + 4ac}/2a \] ?

Answer 12

no

Answer 13

you are not being asked to solve a quadratic equation by finding its roots.
If that was the case you would have something like:\[2x^2-20x+100=0\]and you would be asked to find the roots of this equation. This is where you can either factor or use the quadratic equation.

Answer 14

find the derivative?

Answer 15

What you have in this question is a function \(f\) defined as:\[f=2x^2-20x+100\]and you are being asked to find the value of \(x\) that will minimize this function.

Answer 16

yes - now you are thinking along the right lines

Answer 17

4x - 20 ?

Answer 18

so x = 5

Answer 19

what should the derivative be equal to in order to have a minimum (or maximum) value?

Answer 20

would it be 10