Question

Cho a,b,c>0, a+b+c=3

Chứng minh: a⁵ +b⁵+c⁵+1/a+1/b+1/c> hoặc =6

Answer 1

Theo BĐT AM-GM :

\(a^5+\frac{1}{a}+1+1\ge4\sqrt[4]{a^5\cdot\frac{1}{a}\cdot1\cdot1}=4a\)

Dấu "=" xảy ra \(\Leftrightarrow a^5=\frac{1}{a}=1\Leftrightarrow a=1\)

+ Tương tự :

\(b^5+\frac{1}{b}+1+1\ge4b\) Dấu "=" <=> b = 1

\(c^5+\frac{1}{c}+1+1\ge4c\) Dấu "=" <=> c = 1

Do đó : \(a^5+b^5+c^5+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+6\ge4\left(a+b+c\right)\)

=> đpcm

Dấu "=" <=> a = b = c = 1