Math Is Fun Forum

Jani Wolf

Member

Registered: 2015-09-21

Posts: 2

Solving maximum and minimum problems

I have a question that relates to solving two unknown variables.
Could someone help me out?

: The sum of two numbers is 10. Find the numbers if the sum of their squares is to be a minimum. :

I tried to solve it by writing an equation, then expressing one unkown in terms of the others, substituting, then using differentiation and equate it to '0',
but so far no good.

Thanks in advance. :D

Last edited by Jani Wolf (2015-09-21 07:11:46)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Solving maximum and minimum problems

What kind of numbers are they? Integers? Reals?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

phanthanhtom

Member

Registered: 2012-06-22

Posts: 290

Re: Solving maximum and minimum problems

So a + b = 10.

Thus (a+b)^2 = 100

2(a^2 + b^2) - (a-b)^2 = 100

2(a^2 + b^2) = 100 + (a-b)^2 >= 100

(a^2 + b^2) >= 50.

Equality holds if  (a-b)^2 = 0 or a = b = 5.

It doesn't matter if the numbers are real or integral or whatever.

Jani Wolf

Member

Registered: 2015-09-21

Posts: 2

Re: Solving maximum and minimum problems

The numbers are integers.

Thanks Phanthanhtom, I appreciate the help

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Solving maximum and minimum problems

It doesn't matter if the numbers are real or integral or whatever.

It might not matter to you but when they positive integers the problem can be attacked in other ways.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.