Question

Giải hộ mình mấy câu này với

Chứng minh:

a)\(\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ac}{b}\ge a+b+c\)

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

Answer 1

a)

\(\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ac}{b}\ge a+b+c\\ \Leftrightarrow\dfrac{a^2b^2+b^2c^2+a^2c^2}{abc}\ge\dfrac{\left(a+b+c\right)abc}{abc}\\ \Leftrightarrow a^2b^2+b^2c^2+a^2c^2\ge a^2bc+b^2ac+c^2ab\\ \Leftrightarrow a^2b^2+b^2c^2+a^2c^2-a^2bc-c^2ab-b^2ac\ge0\\ \Leftrightarrow2\left(a^2b^2+b^2c^2+a^2c^2-a^2bc-b^2ac-c^2ab\right)\ge0\\ \Leftrightarrow\left(a^2b^2-2b^2ac+b^2c^2\right)+\left(a^2b^2-2a^2bc+a^2c^2\right)+\left(a^2c^2-2c^2ab+b^2c^2\right)\ge0\\ \Leftrightarrow\left(ab-bc\right)^2+\left(ba-ac\right)^2+\left(ac-ab\right)^2\ge0\left(1\right)\)

Vì BĐT (1) luôn đúng với mọi a,b,c nên \(\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ac}{b}\ge a+b+c\)