Áp dụng BĐT Cô - Si cho các số dương , ta có :
\(\dfrac{ab}{c}+\dfrac{bc}{a}\ge2\sqrt{\dfrac{ab}{c}.\dfrac{bc}{a}}=2\sqrt{b^2}=2b\) ( 1)
\(\dfrac{bc}{a}+\dfrac{ac}{b}\ge2\sqrt{\dfrac{bc}{a}.\dfrac{ac}{b}}=2\sqrt{c^2}=2c\) ( 2)
\(\dfrac{ab}{c}+\dfrac{ac}{b}\ge2\sqrt{\dfrac{ab}{c}.\dfrac{ac}{b}}=2\sqrt{a^2}=2a\) ( 3)
Cộng từng vế của ( 1;2;3) , ta có :
\(\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ac}{b}\ge a+b+c\)
Đẳng thức xảy ra khi : a = b = c
Áp dụng bđt cosi ta có:
\(\dfrac{ab}{c}+\dfrac{bc}{a}\ge2\sqrt{\dfrac{ab}{c}\cdot\dfrac{bc}{a}}=2\sqrt{b^2}=2b\)
Tương tự:
\(\left\{{}\begin{matrix}\dfrac{bc}{a}+\dfrac{ac}{b}\ge2b\\\dfrac{ab}{c}+\dfrac{ac}{b}\ge2a\end{matrix}\right.\)
Cộng 2 vế của các bđt trên ta có:
\(\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{bc}{a}+\dfrac{ac}{b}+\dfrac{ab}{c}+\dfrac{ac}{b}\ge2b+2c+2a\)
\(\Rightarrow2\left(\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ac}{b}\right)\ge2\left(a+b+c\right)\)
\(\Rightarrow\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ac}{b}\ge a+b+c\)
Dấu ''='' xảy ra khi a = b = c
Akai HarumaPhùng Khánh LinhLightning Farron