Question

cho a,b,c > 0 chứng minh a^2/c + b^2/a + c^2/b lớn hơn hoặc bằng a + b + c

Answer 1

Do a, b, c >0

=> a+b+c>0 và \(\dfrac{a^2}{c}\)+\(\dfrac{b^2}{a}\)+\(\dfrac{c^2}{b}\) >0

Áp dụng bất đẳng thức Cô si ta có:

\(\dfrac{a^2}{c}\)+\(\dfrac{b^2}{a}\)+\(\dfrac{c^2}{b}\) \(\ge\) 3 \(\sqrt[3]{\dfrac{a^2b^2c^2}{abc}}\) = 3\(\sqrt[3]{abc}\)

a+b+c \(\ge\) 3 \(\sqrt[3]{abc}\)

=> \(\dfrac{a^2}{c}\)+\(\dfrac{b^2}{a}\)+\(\dfrac{c^2}{b}\) - (a+b+c) \(\ge\) 3\(\sqrt[3]{abc}\) - 3\(\sqrt[3]{abc}\)

=>\(\dfrac{a^2}{c}\)+\(\dfrac{b^2}{a}\)+\(\dfrac{c^2}{b}\)- (a+b+c) \(\ge\) 0

=> \(\dfrac{a^2}{c}\)+\(\dfrac{b^2}{a}\)+\(\dfrac{c^2}{b}\) \(\ge\) a+b+c (dpcm)