Question

CMR:

a/\(a^2+b^2+c^2\ge\text{ab}+bc+c\text{a}\)

b/\(3\left(\text{a}b+bc+c\text{a}\right)\le\left(\text{a}+b+c\right)^2\le3\left(\text{a}^2+b^2+c^2\right)\)

c/\(\text{a}^3+b^3\ge\text{a}b\left(\text{a}+b\right)\)

Answer 1

Answer 2

a) \(a^2+b^2+c^2\ge ab+bc+ac\)

\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2bc-2ac\ge0\)

\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\)( luôn đúng )

Dấu "=" \(\Leftrightarrow a=b=c\)

b) \(\left(a+b+c\right)^2=a^2+b^2+c^2+2ab+2bc+2ac\)

+) vế 1 bđt \(\Leftrightarrow a^2+b^2+c^2\ge ab+bc+ac\)( CMTT câu a )

+) vế 2 bđt \(\Leftrightarrow3a^2+3b^2+3c^2\ge a^2+b^2+c^2+2ab+2bc+2ac\)

\(\Leftrightarrow a^2+b^2+c^2\ge ab+bc+ac\)( CMTT câu a )

Từ đây ta có đpcm

Dấu "=" \(\Leftrightarrow a=b=c\)

c) \(a^3+b^3\ge ab\left(a+b\right)\)

\(\Leftrightarrow\left(a+b\right)\left(a^2-ab+b^2\right)\ge ab\left(a+b\right)\)

\(\Leftrightarrow a^2-ab+b^2\ge ab\)

\(\Leftrightarrow\left(a-b\right)^2\ge0\)( luôn đúng )

Dấu "=" \(\Leftrightarrow a=b\)