Question

cho a, b, c >0 và a+b+c=3.Chứng minh

\(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\ge a^2+b^2+c^2\)

Answer 1

làm xong rồi thì please_sign

áp dụng bđt huyền thoại \(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\ge\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\) =\(\frac{a+b+c}{abc}=\frac{\left(a+b+c\right)^2}{abc\left(a+b+c\right)}\) 

mà \(\left(ab+bc+ac\right)^2\ge3abc\left(a+b+c\right)\) (tụ cm nhé )

\(\Rightarrow\ge\frac{\left(a+b+c^2\right)}{\frac{\left(ab+bc+ac\right)^2}{3}}=\frac{3\left(a+b+c\right)^2\left(a^2+b^2+c^2\right)}{\left(ab+bc+ac\right)^2\left(a^2+b^2+c^2\right)}\)

m,à \(\left(ab+bc+ac\right)^2\left(a^2+b^2+c^2\right)\le\frac{\left(a^2+b^2+c^2+ab+bc+ac+ab+bc+ac\right)^3}{3^3}\)

   =\(\frac{\left(\left(a+b+c\right)^2\right)^3}{27}=27\)

\(\Rightarrow vt\ge\frac{27\left(a^2+b^2+c^2\right)}{27}=a^2+b^2+c^2\)

dau = khi a=b=c=1

Answer 2

hay quá bạn ơi