Finding range of expression based on conditions applied

Brielle James

Brielle James

Answered question

2022-04-06

Finding range of expression based on conditions applied to quadratic expressions
Given that a,b,c are distinct real numbers such that expressions ax2+bx+c,bx2+cx+a and cx2+ax+b are always non-negative. Prove that the quantity a2+b2+c2ab+bc+ca can never lie in (,1][4,).

Answer & Explanation

tralhavahr9c

tralhavahr9c

Beginner2022-04-07Added 16 answers

Step 1
ac=b2+ξ14landab=c2+ξ24landbc=a2+ξ34,ξ1,ξ2,ξ30
We have:
4(a2+b2+c2)a2+b2+c2+ξ1+ξ2+ξ3
So, when ξ1=ξ2=ξ3=0, the fraction is equal to 4 and it can be greater because in that case ξ1+ξ2+ξ3<0 that is a contradiction.
Step 2
Now, we suppose that:
4(a2+b2+c2)a2+b2+c2+ξ1+ξ2+ξ3<1
holds for every a, b, c. This implies:
3(a2+b2+c2)<ξ1+ξ2+ξ3
Substituing, we have: 4(a2+b2+c2)<4(ab+ac+bc)
that is uncorrect because we know that:
a2+b2+c2<4(ab+ac+cb)

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