Mathematics Asked on January 1, 2022
I encountered the following problem in my textbook-
Let $a, b, c$ be three arbitrary real numbers. Denote $$ x = sqrt{b^2-bc+c^2}, y = sqrt{c^2-ca+a^2}, z = sqrt{a^2-ab+b^2} $$ Prove that $$ xy+yz+zx ge a^2+b^2+c^2 $$
Textbook’s Solution:
Rewrite x,y in the following forms $$ x = sqrt{{3c^2over 4}+left(b-{cover 2}right)^2}, y = sqrt{{3c^2over 4}+left(a-{cover 2}right)^2} $$
According to Cauchy-Schwarz inequality, we conclude $$ xy ge {3c^2over 4}+{1over 4}left( 2b-cright)left(2a-c right) $$
which implies
$$ sum_{cyc}xy ge {3over 4}sum_{cyc}c^2 +{1over 4}sum_{cyc}left( 2b-cright)left(2a-c right) = sum_{cyc}a^2. $$
My Approach:
From the given values of $x$ and $y$–
$$ xy = sqrt{left( c^2-bc+b^2right) left(c^2-ca+a^2 right)} ge c^2+csqrt{ab}+ab $$
So, $$xy+yz+zx ge sum_{cyc}a^2+sum_{cyc}asqrt{bc}+sum_{cyc}bc$$
And it rests to prove that-
$$ sum_{cyc}a^2+sum_{cyc}asqrt{bc}+sum_{cyc}bc ge sum_{cyc}a^2 $$
$$ sum_{cyc}asqrt{bc}+sum_{cyc}bc ge 0 $$
Now my question is that how to prove this? Or are we done?
And also if both approachs are correct, what is the equality case?
The equality case is not given in the textbook so I ask.
Thanks!
Your first step is wrong.
Try $a=b=c$.
Id est, you got a right inequality, but after a wrong step and the proof of your last inequality is not relevant already.
Answered by Michael Rozenberg on January 1, 2022
1 Asked on January 25, 2021 by tota
abelian groups cyclic groups group homomorphism group theory monomorphisms
4 Asked on January 25, 2021
1 Asked on January 25, 2021 by michael-blane
abstract algebra ceiling and floor functions combinatorics permutations problem solving
0 Asked on January 25, 2021 by yolbarsop
approximation asymptotics integration riemann sum sequences and series
1 Asked on January 25, 2021 by bellow
9 Asked on January 25, 2021 by user1551
function and relation composition functional equations functions real analysis
2 Asked on January 25, 2021 by cardinal
2 Asked on January 25, 2021 by gene
functional analysis linear algebra matrices norm spectral theory
1 Asked on January 25, 2021 by gal-ben-ayun
0 Asked on January 25, 2021 by jyothi-jain
0 Asked on January 25, 2021 by eyesima
elementary set theory notation proof writing solution verification
6 Asked on January 24, 2021 by user713999
1 Asked on January 24, 2021
1 Asked on January 24, 2021 by scott-frazier
1 Asked on January 24, 2021
1 Asked on January 24, 2021 by methodcl
Get help from others!
Recent Answers
Recent Questions
© 2023 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP