bài này sai đề nha, phải (a+b+c+d)^3 mới đúng
abc+bcd+cda+dab
=ab(c+d)+cd(a+b)≤1/4(a+b)2(c+d)+1/4(c+d)2(a+b)=1/4(a+b)(c+d)(a+b+c+d)≤1/16(a+b+c+d)3
=>16(abc+bcd+cda+dad)<=(a+b+c+d)3
Cho a,b,c,d>0
\(16\left(abc+bcd+cda+dab\right)\le\left(a+b+c+d\right)^3\)
CMBDT
\(ab+bc+cd+da\le\frac{\left(a+b+c+d\right)^2}{4}\)
\(abc+bcd+cda+dab\le\frac{\left(a+b+c+d\right)^3}{16}\)
CHỨNG MINH \(abc+bcd+cda+dab\le\frac{1}{16}\left(a+b+c+d\right)^3\)
cho abc + bcd + cda + dab = a+b+c+d+\(\sqrt{2015}\)
CMR : \(\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)\left(d^2+1\right)\ge2015\)
Cho a,b,c,d là các số thực thoả mãn điều kiện
\(abc+bcd+cda+dab=a+b+c+d+\sqrt{2012}\)
CMR: \(\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)\left(d^2+1\right)\ge2012\)
Cho a, b, c, d là các số thực thỏa mãn điều kiện \(abc+bcd+cda+dab=a+b+c+d+\sqrt{2016}\)
Chứng minh rằng: \(\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)\left(d^2+1\right)\ge2016\)
cho a,b,c,d là cá số thực tm đk
\(abc+bcd+cda+dab=a+b+c+d\)\(+\sqrt{2012}\)
cmr \(\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)\left(d^2+1\right)\ge2012\)
Chứng minh giúp mình mấy câu bất đẳng thức này nha
a) \(\frac{2\sqrt{ab}}{\sqrt{a}+\sqrt{b}}\le\sqrt[4]{ab}\left(a,b>0\right)\)
b) \(\left(\sqrt{a}+\sqrt{b}\right)^8\ge64ab\left(a+b\right)^2\left(a,b>0\right)\)
c) \(y\left(\frac{1}{x}+\frac{1}{x}\right)+\frac{1}{y}\left(x+z\right)\le\left(\frac{1}{x}+\frac{1}{z}\right)\left(x+z\right)\left(0< x\le y\le z\right)\)
d) \(a+b+c\ge3\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\left(a,b,c>0;a+b+c=abc\right)\)
ta có:
(a+b)(b+c)(c+a)=(a+b+c)(ab+bc+ca)-abc\(\ge\left(a+b+c\right)\left(ab+bc+ca\right)-\frac{1}{9}\left(a+b+c\right)\left(ab+bc+ca\right)=\frac{8}{9}\left(a+b+c\right)\left(ab+bc+ca\right)\)
\(\frac{x}{x+yz}+\frac{y}{y+zx}+\frac{z}{z+xy}=\frac{x}{\left(x+y\right)\left(x+z\right)}+\frac{y}{\left(y+x\right)\left(y+z\right)}+\frac{z}{\left(z+x\right)\left(z+y\right)}=\frac{2\left(xy+yz+zx\right)}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}\le\frac{9}{4\left(xy+yz+zx\right)}=\frac{9}{4}\)
