Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
\(VT=\sqrt{\frac{ab}{\left(a+c\right)\left(b+c\right)}}+\sqrt{\frac{bc}{\left(a+b\right)\left(a+c\right)}}+\sqrt{\frac{ca}{\left(b+c\right)\left(a+b\right)}}\)
Áp dụng bất đẳng thức Cauchy - Schwarz
\(\Rightarrow\sqrt{\frac{ab}{\left(a+c\right)\left(b+c\right)}}\le\frac{\frac{a}{a+c}+\frac{b}{b+c}}{2}\)
Tượng tự ta có \(\hept{\begin{cases}\sqrt{\frac{bc}{\left(a+b\right)\left(a+c\right)}}\le\frac{\frac{b}{a+b}+\frac{c}{a+c}}{2}\\\sqrt{\frac{ca}{\left(b+c\right)\left(a+b\right)}}\le\frac{\frac{c}{b+c}+\frac{a}{a+b}}{2}\end{cases}}\)
\(\Rightarrow VT\le\frac{\left(\frac{a}{a+b}+\frac{b}{a+b}\right)+\left(\frac{c}{a+c}+\frac{a}{c+a}\right)+\left(\frac{c}{b+c}+\frac{b}{c+b}\right)}{2}\)
\(\Rightarrow VT\le\frac{\frac{a+b}{a+b}+\frac{c+a}{c+a}+\frac{b+c}{b+c}}{2}=\frac{3}{2}\) ( đpcm )
Dấu " = " xảy ra khi \(a=b=c=\frac{1}{3}\)
1,
\(\frac{a}{1+\frac{b}{a}}+\frac{b}{1+\frac{c}{b}}+\frac{c}{1+\frac{a}{c}}=\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\ge\frac{\left(a+b+c\right)^2}{2\left(a+b+c\right)}=\frac{a+b+c}{2}\ge\frac{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}{2}=\frac{2}{2}=1\left(Q.E.D\right)\)
1) c/m \(a+b+c\ge\sqrt{ab}+\sqrt{bc}+\sqrt{ac}\)
áp dụng BĐT cô shi cho 2 số thực dương ta có:
\(a+b\ge2\sqrt{ab}\);\(b+c\ge2\sqrt{bc}\);\(a+c\ge2\sqrt{ac}\)
cộng vế vs vế:\(2\left(a+b+c\right)\ge2\left(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\right)\)
↔\(a+b+c\ge\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\)
dấu = xảy ra khi a=b=c
vậy...
b)ta có:
\(\frac{1}{\sqrt{1}}>\frac{1}{\sqrt{2}}>\frac{1}{\sqrt{3}}>...>\frac{1}{\sqrt{25}}\)→\(A>\frac{1}{\sqrt{25}}+\frac{1}{\sqrt{25}}+...+\frac{1}{\sqrt{25}}\)(25 số hạng)
\(A>\frac{25}{\sqrt{25}}=\sqrt{25}=5\)
vậy.....
\(\sum\dfrac{a}{\sqrt{ab+b^2}}=\sum\dfrac{a\sqrt{2}}{\sqrt{2b\left(a+b\right)}}\ge\sum\dfrac{2\sqrt{2}a}{2b+a+b}=2\sqrt{2}\sum\dfrac{a}{a+3b}\)
\(=2\sqrt{2}\sum\dfrac{a^2}{a^2+3ab}\ge\dfrac{2\sqrt{2}\left(a+b+c\right)^2}{a^2+b^2+c^2+3\left(ab+bc+ca\right)}\)
\(=\dfrac{2\sqrt{2}\left(a+b+c\right)^2}{\left(a+b+c\right)^2+ab+bc+ca}\ge\dfrac{2\sqrt{2}\left(a+b+c\right)^2}{\left(a+b+c\right)^2+\dfrac{1}{3}\left(a+b+c\right)^2}=\dfrac{3\sqrt{2}}{2}\)
Cho a,b,c là các số thực dương thỏa mãn a+b+c = 3
Chứng minh rằng với mọi k > 0 ta luôn có....
.
Cho a,b,c là các số thực dương thỏa mãn a+b+c = 3
Chứng minh rằng với mọi k > 0 ta luôn có
BĐT cần c/m tương đương:
\(2\left(a^3+b^3+c^3+d^3\right)\ge2+\dfrac{3}{2}\sqrt{4+2\left(ab+ac+ad+bc+bd+cd\right)}\)
\(\Leftrightarrow2\left(a^3+b^3+c^3+d^3\right)\ge2+\dfrac{3}{2}\sqrt{\left(a+b+c+d\right)^2}\)
\(\Leftrightarrow2\left(a^3+b^3+c^3+d^3\right)\ge2+\dfrac{3}{2}\left(a+b+c+d\right)\)
\(\Leftrightarrow4\left(a^3+b^3+c^3+d^3\right)\ge4+3\left(a+b+c+d\right)\)
Dễ dàng chứng minh điều này bằng AM-GM:
\(a^3+a^3+1+b^3+b^3+1+c^3+c^3+1+d^3+d^3+1\ge3a^2+3b^2+3c^2+3d^2\)
\(\Rightarrow2\left(a^3+b^3+c^3+d^3\right)+4\ge12\)
\(\Rightarrow a^3+b^3+c^3+d^3\ge4\) (1)
Lại có:
\(a^2+b^2+c^2+d^2\ge\dfrac{1}{4}\left(a+b+c+d\right)^2\)
\(\Rightarrow a+b+c+d\le4\) (2)
(1);(2) \(\Rightarrow4\left(a^3+b^3+c^3+d^3\right)\ge16\ge4+3.4\ge4+3\left(a+b+c+d\right)\) (đpcm)
\(\sqrt{\dfrac{ab+2c^2}{1+ab-c^2}}=\sqrt{\dfrac{ab+2c^2}{a^2+b^2+ab}}\)\(=\dfrac{ab+2c^2}{\sqrt{\left(a^2+b^2+ab\right)\left(ab+c^2+c^2\right)}}\)\(\ge\dfrac{2\left(ab+2c^2\right)}{a^2+b^2+2ab+2c^2}\)\(\ge\dfrac{2\left(ab+2c^2\right)}{2\left(a^2+b^2\right)+2c^2}\)\(=\dfrac{ab+2c^2}{a^2+b^2+c^2}\)
\(\Rightarrow\sqrt{\dfrac{ab+2c^2}{1+ab-c^2}}\ge ab+2c^2\)
Tương tự: \(\sqrt{\dfrac{bc+2a^2}{1+bc-a^2}}\ge bc+2a^2\); \(\sqrt{\dfrac{ac+2b^2}{1+ac-b^2}}\ge ac+2b^2\)
Cộng vế với vế \(\Rightarrow VT\ge2a^2+2b^2+2c^2+ab+bc+ac=2+ab+bc+ac\)
Dấu = xảy ra khi \(a=b=c=\dfrac{1}{\sqrt{3}}\)
