Question

CMR:với mọi số thực dương a;b;c ta có:

\(\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2}\ge\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\)

Answer 1

i don't no

Answer 2

Áp dụng bđt Cosi cho 2 số dương, ta có:

* ​\(\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2}=\frac{a^3}{b^2}+a+\frac{b^3}{c^2}+b+\frac{c^3}{a^2}+c-a-b-c\)\(\ge2\sqrt{\frac{a^3}{b^2}.a}+2\sqrt{\frac{b^3}{c^2}.b}+2\sqrt{\frac{c^3}{a^2}.c}-a-b-c\)\(=2.\frac{a^2}{b}+2.\frac{b^2}{c}+2.\frac{c^2}{a}-a-b-c\)

* \(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}-a-b-c=\frac{a^2}{b}+b+\frac{b^2}{c}+c+\frac{c^2}{a}+a-2a-2b-2c\)

\(\ge2\sqrt{\frac{a^2}{b}.b}+2\sqrt{\frac{b^2}{c}.c}+2\sqrt{\frac{c^2}{a}.a}-2a-2b-2c=0\)

\(\Rightarrow\)\(2.\frac{a^2}{b}+2.\frac{b^2}{c}+2.\frac{c^2}{a}-a-b-c\ge\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\)

\(\Rightarrow\)\(\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2}\ge\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\)

Nếu đúng cho mình nhé.