\(A=\sqrt{c\left(a-c\right)}+\sqrt{c\left(b-c\right)}\)
\(\Rightarrow A^2=\left(\sqrt{c\left(a-c\right)}+\sqrt{c\left(b-c\right)}\right)^2\)\(=\left(\sqrt{c}.\sqrt{a-c}+\sqrt{c}.\sqrt{b-c}\right)^2\)
\(\Rightarrow A^2\le\left(c+b-c\right)\left(c+ a-c\right)\left(\text{ Bunhiacopxki}\right)\)
\(\Rightarrow A^2\le ab\Leftrightarrow A\le\sqrt{ab}\left(đpcm\right)\)
\(\)
1. Cho a,b,c,d là các số dương. Chứng minh rằng: \(\sqrt{ab}+\sqrt{cd}\le\sqrt{\left(a+d\right)\left(b+c\right)}\)
2. Cho (x;y;z) và (a;b;c) là các số dương. Chứng minh rằng: \(\sqrt[3]{abc}+\sqrt[3]{xyz}\le\sqrt[3]{\left(a+x\right)\left(b+y\right)\left(c+z\right)}\)
3. Cho \(c>0\) và \(a,b\ge c\). Chứng minh rằng: \(\sqrt{c\left(a-c\right)}+\sqrt{c\left(b-c\right)}\le\sqrt{ab}\)
Cho a, b, c >0. CMR: \(\dfrac{a+b+c}{3}\) - \(\sqrt[3]{abc}\) ≤ \(\dfrac{\left(\sqrt{a}-\sqrt{b}\right)^2+\left(\sqrt{b}-\sqrt{c}\right)^2+\left(\sqrt{c}-\sqrt{a}\right)^2}{3}\)
1. cho \(0< a\le b\le c\) . Cmr: \(\frac{2a^2}{b^2+c^2}+\frac{2b^2}{c^2+a^2}+\frac{2c^2}{a^2+b^2}\le\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\)
2. cho \(a,b,c\ge0\). cmr: \(a^2+b^2+c^2+3\sqrt[3]{\left(abc\right)^2}\ge2\left(ab+bc+ca\right)\)
3. \(a,b,c>0.\) Cmr: \(\sqrt{\left(a^2b+b^2c+c^2a\right)\left(ab^2+bc^2+ca^2\right)}\ge abc+\sqrt[3]{\left(a^3+abc\right)\left(b^3+abc\right)\left(c^3+abc\right)}\)
4. \(a,b,c>0\). Tìm Min \(P=\left(\frac{a}{a+b}\right)^4+\left(\frac{b}{b+c}\right)^4+\left(\frac{c}{c+a}\right)^4\)
a)Cho 0 < c ; c < b ; b < a . CMR:\(\sqrt{c\left(a-c\right)}+\sqrt{b\left(b-c\right)}\le\sqrt{ab}\)
b)Cho \(x\ge1;y\ge1\). CMR:\(\dfrac{1}{1+x^2}+\dfrac{1}{1+y^2}\ge\dfrac{2}{1+xy}\)
Cho a,b,c,d >0. Chứng minh rằng:
\(\sqrt{ab}+\sqrt{cd}\le\sqrt{\left(a+d\right)\left(b+c\right)}\)
Bài 1 : Cho a>c , b>c ( a,b,c>0). Cmr : \(\sqrt{c\sqrt{a-c}}+\sqrt{c\sqrt{b-c}}\le\sqrt{ab}\) (Hướng dẫn : chia cả 2 vế cho \(\sqrt{ab}\) , dùng cô-si)
Bài 2 : Cho \(a\ge1;b\ge1\) . Cmr \(a\sqrt{b-1}+b\sqrt{a-1}\le ab\)
Bài 3 : Tìm GTNN của \(A=\left(a+1\right)^2+\left(\frac{a^2}{a+1}+2\right)^2\) với mọi a\(\ne1\)
cho bốn số thực dương a,b,c,d chứng minh rằng \(\sqrt{c\left(a-c\right)}+\sqrt{c\left(b-c\le\right)}\le\sqrt{ab}\)
a) cmr (ax+by+cz)\(^2\)≤\(\left(a^2+b^2+c^2\right)\left(x^2+y^2+z^2\right)\)
b) cho a,b,c >0 tm \(a^2+b^2+c^2=1\)
cmr :\(\frac{1}{\sqrt{a^2+1}}+\frac{1}{\sqrt{b^2+1}}+\frac{1}{\sqrt{c^2+1}}\le\frac{a}{2\left(a+b+c\right)}\)
1. a) \(\left\{{}\begin{matrix}x,y,z>0\\xyz=1\end{matrix}\right.\). Tìm max \(P=\frac{1}{\sqrt{x^5-x^2+3xy+6}}+\frac{1}{\sqrt{y^5-y^2+3yz+6}}+\frac{1}{\sqrt{z^5-z^2+zx+6}}\)
b) \(\left\{{}\begin{matrix}x,y,z>0\\xyz=8\end{matrix}\right.\). Min \(P=\frac{x^2}{\sqrt{\left(1+x^3\right)\left(1+y^3\right)}}+\frac{y^2}{\sqrt{\left(1+y^3\right)\left(1+z^3\right)}}+\frac{z^2}{\sqrt{\left(1+z^3\right)\left(1+x^3\right)}}\)
c) \(x,y,z>0.\) Min \(P=\sqrt{\frac{x^3}{x^3+\left(y+z\right)^3}}+\sqrt{\frac{y^3}{y^3+\left(z+x\right)^3}}+\sqrt{\frac{z^3}{z^3+\left(x+y\right)^3}}\)
d) \(a,b,c>0;a^2+b^2+c^2+abc=4.Cmr:2a+b+c\le\frac{9}{2}\)
e) \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c=3\end{matrix}\right.\). Cmr: \(\frac{a}{b^3+ab}+\frac{b}{c^3+bc}+\frac{c}{a^3+ca}\ge\frac{3}{2}\)
f) \(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca+abc=4\end{matrix}\right.\) Cmr: \(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\le3\)
g) \(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca+abc=2\end{matrix}\right.\) Max : \(Q=\frac{a+1}{a^2+2a+2}+\frac{b+1}{b^2+2b+2}+\frac{c+1}{c^2+2c+2}\)
