Question

Find 2 numbers a and b such that (a-b)^2 < (a+b)(a-b)< (a+b)^2 Find two numbes a and b such that this is not true

Answer 1

Suppose for the moment that the inequality holds for all
`a,b`:

`(a-b)^2<(a+b)(a-b)<(a+b)^2`

Expanding everything gives

`a^2-2ab+b^2<a^2-b^2<a^2+2ab+b^2`

`\implies-ab<-b^2<ab`

In particular, the inequality says that
`-ab<ab` for any choice of
`a,b`. But if
`a<0` and
`b>0`, then
`-ab>0` while
`ab<0`.

So one possible choice of
`a,b` could be
`a=-1` and
`b=1`. Then we get

`(-1-1)^2=4`

`(-1+1)(-1-1)=2`

`(-1+1)^2=0`

but clearly it's not true that
`4<2<0`.