Question

Cho a , b , c thỏa mãn a > c , b > c > 0 . Chứng minh rằng :

\(\sqrt{c\left(a-c\right)}+\sqrt{c\left(b-c\right)}\le\sqrt{ab}\)

Answer 1

Áp dụng bất đẳng thức cô - si cho 2 số không âm ta có :

\(\sqrt{\dfrac{c\left(a-c\right)}{ab}}+\sqrt{\dfrac{c\left(b-c\right)}{ab}}\le\dfrac{1}{2}\left(\dfrac{c}{b}+\dfrac{a-c}{a}\right)+\dfrac{1}{2}\left(\dfrac{c}{a}+\dfrac{b-c}{b}\right)\)

\(\Rightarrow\dfrac{\sqrt{c\left(a-c\right)}}{\sqrt{ab}}+\dfrac{\sqrt{c\left(b-c\right)}}{\sqrt{ab}}\le1\)

\(\Rightarrow\sqrt{c\left(a-c\right)}+\sqrt{c\left(b-c\right)}\le\sqrt{ab}\left(đpcm\right)\)