Cho x, y, z > 0 thỏa mãn: \(x^2+y^2+z^2=3\). Chứng minh: \(\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}\ge3\).
Cho x,y,z > 0 thỏa mãn xy + yz +zx = 1.Chứng minh
\(\frac{x-y}{z^2+1}\)+\(\frac{y-z}{x^2+1}\)+\(\frac{z-x}{y^2+1}\)=0
\(\dfrac{x-y}{z^2+1}=\dfrac{x-y}{z^2+xy+yz+zx}=\dfrac{x-y}{z\left(z+y\right)+x\left(z+y\right)}=\dfrac{x-y}{\left(x+z\right)\left(z+y\right)}\)
Tương tự: \(\dfrac{y-z}{x^2+1}=\dfrac{y-z}{\left(x+y\right)\left(x+z\right)}\);\(\dfrac{z-x}{y^2+1}=\dfrac{z-x}{\left(x+y\right)\left(y+z\right)}\)
Cộng vế với vế \(\Rightarrow VT=\dfrac{x-y}{\left(x+z\right)\left(y+z\right)}+\dfrac{y-z}{\left(x+y\right)\left(x+z\right)}+\dfrac{z-x}{\left(x+y\right)\left(y+z\right)}\)
\(=\dfrac{\left(x-y\right)\left(x+y\right)+\left(y-z\right)\left(y+z\right)+\left(z-x\right)\left(z+x\right)}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}\)
\(=\dfrac{x^2-y^2+y^2-z^2+z^2-x^2}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}=0\)(đpcm)
Cho x,y,z>0 thỏa mãn xy+yz+zx=1. Chứng minh \(\frac{x}{x^2-yz+3}+\frac{y}{y^2-zx+3}+\frac{z}{z^2-xy+3}\ge\frac{1}{x+y+z}\)
Cho \(\hept{\begin{cases}x,y,z>0\\xy+yz+zx=1\end{cases}}\). Chứng minh rằng:
\(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}\ge3+\sqrt{\frac{\left(x+y\right)\left(x+z\right)}{x^2}}+\sqrt{\frac{\left(y+z\right)\left(y+x\right)}{y^2}}+\sqrt{\frac{\left(z+x\right)\left(z+y\right)}{z^2}}\)
1111111111111111111
\(VT=\Sigma\frac{xy+yz+zx}{xy}=3+\Sigma\frac{z\left(x+y\right)}{xy}\)
Đến đây để ý \(\frac{1}{2}\left[\frac{z\left(x+y\right)}{xy}+\frac{y\left(z+x\right)}{zx}\right]\ge\sqrt{\frac{\left(z+x\right)\left(x+y\right)}{x^2}}\left(\text{AM - GM}\right)\)
Là xong.
Cho x, y, z > 0 thỏa mãn xyz = 1. Chứng minh :
\(\frac{xy}{x^5+xy+y^5}+\frac{yz}{y^5+yz+z^5}+\frac{zx}{z^5+zx+x^5}\le1\)
ủa đây là toám lớp 1 hả anh
Forever_Alone tên là Anh nhưng ko bt họ
Cho x,y,z là các số dương thỏa mãn x2+y2+z2=3. Chứng minh rằng
\(\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}\ge3\)
Đặt vế trái là P
\(P=\frac{x^2y^2+y^2z^2+z^2x^2}{xyz}\Rightarrow P^2=\frac{x^4y^4+y^4z^4+z^4x^4+2x^2y^2z^2\left(x^2+y^2+z^2\right)}{x^2y^2z^2}\)
\(P^2\ge\frac{x^2y^2z^2\left(x^2+y^2+z^2\right)+6x^2y^2z^2}{x^2y^2z^2}=\frac{9x^2y^2z^2}{x^2y^2z^2}=9\)
\(\Rightarrow P\ge3\)
Dấu "=" xảy ra khi \(x=y=1=z\)
x+y+z=3 ms lm đc
Cho x,y,z > 0 thỏa mãn xy + yz +zx = 1.Chứng minh
\(\frac{x-y}{z^2+1}\)+\(\frac{y-z}{x^2+1}\)+\(\frac{z-x}{y^2+1}\)=0
Vì xy + yz + zx = 1 ta có :
\(\frac{x-y}{z^2+1}+\frac{y-z}{x^2+1}+\frac{z-x}{y^2+1}=\frac{x-y}{z^2+xy+yz+zx}+\frac{y-z}{x^2+xy+yz+zx}+\frac{z-x}{y^2+xy+yz+zx}\)
\(=\frac{x-y}{\left(y+z\right)\left(z+x\right)}+\frac{y-z}{\left(x+y\right)\left(x+z\right)}+\frac{z-x}{\left(y+z\right)\left(x+y\right)}\)
\(=\frac{\left(x-y\right)\left(x+y\right)+\left(y-z\right)\left(y+z\right)+\left(x+z\right)\left(z-x\right)}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}\)
\(=\frac{x^2-y^2+y^2-z^2+z^2-x^2}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}=\frac{0}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}=0\)(ĐPCM)
Cho x,y,z thỏa mãn x+y+z=\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\). Chứng minh rằng
\(\frac{1}{\left(2xy+yz+zx\right)^2}+\frac{1}{\left(2yz+zx+xy\right)^2}+\frac{1}{\left(2xz+xy+yz\right)^2}\le\frac{3}{16x^2y^2z^2}\)
cho x;y;z>0 thỏa mãn x+y+z=3.CMR:\(\frac{x}{x+yz}+\frac{y}{y+zx}+\frac{z}{z+xy}\ge\frac{3}{2}\)
ta caàn chứng minh bđt
\(\frac{x}{x+yz}+\frac{y}{y+zx}\ge\frac{x}{x+xz}+\frac{y}{y+yz}=\frac{1}{1+z}+\frac{1}{1+z}=\frac{2}{1+z}\)
tương tự + vào, dùng svác sơ
Cho 3 số thực dương x , y , z thỏa mãn \(x+y+z\ge3\)
Chứng minh rằng: \(\frac{x^2}{x+\sqrt{yz}}+\frac{y^2}{y+\sqrt{xz}}+\frac{z^2}{z+\sqrt{xy}}\ge\frac{3}{2}\)
Áp dụng BĐT Cô-si dạng Engel,ta có :
\(\frac{x^2}{x+\sqrt{yz}}+\frac{y^2}{y+\sqrt{xz}}+\frac{z^2}{z+\sqrt{xy}}\ge\frac{\left(x+y+z\right)^2}{x+y+z+\sqrt{xy}+\sqrt{yz}+\sqrt{xz}}\)
Mà \(\sqrt{xy}+\sqrt{yz}+\sqrt{xz}\le x+y+z\)
\(\Rightarrow\)\(\frac{\left(x+y+z\right)^2}{x+y+z+\sqrt{xy}+\sqrt{yz}+\sqrt{xz}}\ge\frac{\left(x+y+z\right)^2}{2\left(x+y+z\right)}=\frac{x+y+z}{2}\ge\frac{3}{2}\)
Dấu "=" xảy ra khi x = y = z = \(\frac{3}{2}\)
nhầm sửa x = y = z = 1 nha
Áp dụng bất đẳng thức cộng mẫu số
\(\Rightarrow\frac{x^2}{x+\sqrt{yz}}+\frac{y^2}{y+\sqrt{xz}}+\frac{z^2}{z+\sqrt{xy}}\ge\frac{\left(x+y+z\right)^2}{x+y+z+\sqrt{yz}+\sqrt{xz}+\sqrt{xy}}\)
Xét \(\frac{\left(x+y+z\right)^2}{x+y+z+\sqrt{yz}+\sqrt{xz}+\sqrt{xy}}\)
Áp dụng bất đẳng thức Cacuchy cho 2 bộ số thực không âm
\(\Rightarrow\hept{\begin{cases}\sqrt{yz}\le\frac{y+z}{2}\\\sqrt{xz}\le\frac{x+z}{2}\\\sqrt{xy}\le\frac{x+y}{2}\end{cases}}\)
\(\Rightarrow\sqrt{yz}+\sqrt{xz}+\sqrt{xy}\le\frac{y+z}{2}+\frac{x+z}{2}+\frac{x+y}{2}\)
\(\Rightarrow\sqrt{yz}+\sqrt{xz}+\sqrt{xy}\le\frac{2\left(x+y+z\right)}{2}\)
\(\Rightarrow\sqrt{yz}+\sqrt{xz}+\sqrt{xy}\le x+y+z\)
\(\Rightarrow x+y+z+\sqrt{xy}+\sqrt{yz}+\sqrt{xz}\le2\left(x+y+z\right)\)
\(\Rightarrow\frac{\left(x+y+z\right)^2}{x+y+z+\sqrt{xy}+\sqrt{xz}+\sqrt{yz}}\ge\frac{\left(x+y+z\right)^2}{2\left(x+y+z\right)}=\frac{x+y+z}{2}\)
Ta có : \(x+y+z\ge3\)
\(\Rightarrow\frac{x+y+z}{2}\ge\frac{3}{2}\)
\(\Rightarrow\frac{\left(x+y+z\right)^2}{x+y+z+\sqrt{xy}+\sqrt{xz}+\sqrt{yz}}\ge\frac{3}{2}\)
Vì \(\frac{x^2}{x+\sqrt{yz}}+\frac{y^2}{y+\sqrt{xz}}+\frac{z^2}{z+\sqrt{xy}}\ge\frac{\left(x+y+z\right)^2}{x+y+z+\sqrt{yz}+\sqrt{xz}+\sqrt{xy}}\)
\(\Rightarrow\frac{x^2}{x+\sqrt{yz}}+\frac{y^2}{y+\sqrt{xz}}+\frac{z^2}{z+\sqrt{xy}}\ge\frac{3}{2}\left(đpcm\right)\)
Chúc bạn học tốt !!!