Question

Cm: (-a+b+c)/2a + (a-b+c)/2b + (a+b-c)/2 cho a,b,c >0

Answer 1

chứng minh:\(P=\dfrac{-a+b+c}{2a}+\dfrac{a-b+c}{2b}+\dfrac{a+b-c}{2c}\ge\dfrac{3}{2}\)

\(P=\dfrac{1}{2}\left(\dfrac{b}{a}+\dfrac{c}{a}+\dfrac{a}{b}+\dfrac{c}{b}+\dfrac{a}{c}+\dfrac{b}{c}-3\right)=\dfrac{1}{2}\left(\dfrac{a}{b}+\dfrac{b}{a}\right)+\dfrac{1}{2}\left(\dfrac{b}{c}+\dfrac{c}{b}\right)+\dfrac{1}{2}\left(\dfrac{a}{c}+\dfrac{c}{a}\right)-\dfrac{3}{2}\)Áp dụng BĐT cauchy:

\(\dfrac{a}{b}+\dfrac{b}{a}\ge2\sqrt{\dfrac{a}{b}.\dfrac{b}{a}}=2\)

tương tự với các phân thức còn lại ta có đpcm.

dấu = xảy ra khi a=b=c