Given that `a ,b ,c` are distinct real numbers such that expressions `a x^2+b x+c ,b x^2+c x+aa n dc x^2+a x+b` are always non-negative. Prove that the quantity `(a^2+b^2+c^2)//(a b+b c+c a)` can never lie inn `(-oo,1)` .

Explanation

Given
`a ne b ne c, a, b, c in ` R . Now,
`ax^(2) + bx + c ge 0`
`rArr b^(2) - 4 ac le 0 and a gt 0` (1)
`bx^(2) + cx + a ge 0`
`rArr c^(2) - 4ab le 0 and b gt 0` (2)
`cx^(2) + ax + b ge 0`
`rArr a^(2) - 4bc le 0 and c gt 0 ` (3)
Equality cannot hold simulteneously in (1), (2), and (3).
`therefore a^(2) + b^(2) + c^(2) - 4ac - 4bc - 4ac lt 0`
or `(a^(2)+b^(2) +c^(2))/(ab + bc + ca) gt 4` (4)
Now, `(a-b)^(2) + (b -c)^(2) + (c -a)^(2) gt 0 ( therefore ` a, b, c are distinct)
`rArr (a^(2)+b^(2) +c^(2))/(ab + bc + ca) gt 1`
From (4) and (5),` (a^(2)+ b^(2) + c^(2) //(ab + bc + ca)` can never lie in `(-infty, 1] cup [4, infty ).`