# If A2+B2+C2 Ab Bc Ca=0 Then Prove That A=B=C

Given that \$a\$, \$b\$, \$c\$ are non-negative real numbers such that \$a+b+c=3\$, how can we prove that:

\$a^2+b^2+c^2+ab+bc+cage6\$

By squaring \$a+b+c=3\$ we get\$\$(a+b+c)^2=a^2+b^2+c^2+2(ab+ac+bc)=9.\$\$

From the AM-GM inequality (or from the fact that \$(x-y)^2=x^2+y^2-2xyge 0\$, i.e. \$2xyle x^2+y^2\$)we have\$\$ab+ac+bc le fraca^2+b^22+fraca^2+c^22+fracb^2+c^22=a^2+b^2+c^2,\$\$i.e. \$frac12(a^2+b^2+c^2) ge frac12(ab+ac+bc)\$, which is equivalent to\$frac12(a^2+b^2+c^2) - frac12(ab+ac+bc) ge0\$.

Bạn đang xem: If a2+b2+c2 ab bc ca=0 then prove that a=b=c

By adding the above equality & inequality together you get\$\$frac32(a^2+b^2+c^2+ab+ac+bc)ge9,\$\$which is equivalent to\$\$a^2+b^2+c^2+ab+ac+bcge6.\$\$

By the Cauchy-Schwarz inequality,\$\$6=1*(a+b)+1*(b+c)+1*(a+c)leqsqrt1^2+1^2+1^2sqrt(a+b)^2+(b+c)^2+(a+c)^2\$\$In other words\$\$(a+b)^2+(b+c)^2+(a+c)^2geq 12\$\$which is the same as the desired inequality.

If \$x,y,z\$ are nonnegative reals, then\$x^2+y^2+z^2ge xy+yz+zx\$ (with equality iff \$x=y=z\$), hence\$3(x^2+y^2+z^2)ge x(x+y+z)+y(y+z+x)+z(z+x+y) = (x+y+z)^2\$(with equality iff \$x=y=z\$).Letting \$x=a+b, y=b+c, z=a+c\$, we find \$x+y+z=6\$ and\$3(a+b)^2+3(b+c)^2+3(c+a)^2 ge 36\$.Note that \$(a+b)^2+(b+c)^2+(c+a)^2= 2(a^2+b^2+c^2+ab+bc+ca)\$ so that we actually showed\$\$ a^2+b^2+c^2+ab+bc+cage 6\$\$with equality iff \$a=b=c\$.

Xem thêm: Vitamin 3B Có Tác Dụng Gì ? Vitamin 3B Có Tác Dụng Gì Cho Da

\$\$ a^2 + b^2 + c^2 + ab + bc + ac = (a+b+c)^2 - (ab + bc + ac) = 9 - (ab + bc + ac)\$\$Now it remains lớn show that max value of \$(ab + bc + ac)\$ is \$3\$. For that, we know the AM-GM equality ( for \$a,b, c >0\$ ) that \$3(a^2 + b^2 + c^2) geq (a+ b + c)^2 geq 3(ab +bc +ac)\$. From the last two part we have \$(a+ b + c)^2 geq 3(ab +bc +ac) implies 9 geq 3 (ab +bc +ac) implies 3 geq ab +bc +ac\$

Hence we have \$\$ a^2 + b^2 + c^2 + ab + bc + ac = 9 - (ab + bc + ac) geq 9 - 3 = 6\$\$

Thanks for contributing an answer to lớn ccevents.vnematics Stack Exchange!

Please be sure lớn answer the question. Provide details & share your research!

But avoid

Asking for help, clarification, or responding to other answers.Making statements based on opinion; back them up with references or personal experience.

Use ccevents.vnJax khổng lồ format equations. ccevents.vnJax reference.

Xem thêm: Top 10+ Cách Làm Thế Nào Để Hết Thâm Mắt Lâu Năm Tại Nhà Hiệu Quả

To learn more, see our tips on writing great answers.

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Site thiết kế / logo sản phẩm © 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Rev2022.12.14.43120

Your privacy

By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.