Question

If ab + bc + ca is equal or greater than 2, what is the minimum possible value for a^2 + b^2 + c^2? This was on a noncalculator paper

Answer 1

2

Explanation:

We start with the identity

`(a-b)^2+(b-c)^2+(c-a)^2`
`qquad = 2(a^2+b^2+c^2)-2(ab+bc+ca)`

This shows that

`2(a^2+b^2+c^2)-2(ab+bc+ca) >= 0`
(since a sum of squares of real numbers can not be negative)

Thus
`a^2+b^2+c^2 >= ab+bc+ca >= 2`

Thus the minimum possible value for `a^2+b^2+c^2` is `2` (this minimum can be achieved if `a=b=c=sqrt{2/3}`)