Question

CM bất đẳng thức sau:

a, Cho a>c , b>c , c>0

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

b, CM

\(\dfrac{2005}{\sqrt{2006}}+\dfrac{2006}{\sqrt{2005}}>\sqrt{2005}+\sqrt{2006}\)

help me!!

Answer 1

a) vì ab > 0 nên chia cả hai vế Bất đẳng thức cho \(\sqrt{ab}\) ta được

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

Áp dụng Bất đẳng thức Cauchy cho hai số

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

vậy nên ta có đpcm

Answer 2

\(\frac{2005}{\sqrt{2006} }+\frac{2006}{\sqrt{2005} }>\sqrt{2005}+\sqrt{2006} \)

<=>\(2005\sqrt{2005}+2006\sqrt{2006}>2005\sqrt{2006}+2006\sqrt{2005} \)

<=>\(\sqrt{2006}<\sqrt{2005} \)