Sao không nói x , y , z thuộc N cho nhanh bạn
Sao không nói x , y , z thuộc N cho nhanh bạn
Cho x,y,z>0 và xy+yz+zx=1
a, tính giá trị biểu thức:
\(P=x\sqrt{\frac{\left(1+y^2\right)\left(1+z^2\right)}{1+x^2}}+y\sqrt{\frac{\left(1+x^2\right)\left(1+z^2\right)}{1+y^2}}+z\sqrt{\frac{\left(1+x^2\right)\left(1+y^2\right)}{1+z^2}}\)
b, CMR:
\(\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}=\frac{2xy}{\sqrt{\left(1+x^2\right)\left(1+y^2\right)\left(1+z^2\right)}}\)
Cho 3 số thực x,y,z đôi một phân biệt sao cho
\(\left(y-z\right)\sqrt[3]{1-x^3}+\left(z-x\right)\sqrt[3]{1-y^3}+\left(x-y\right)\sqrt[3]{1-z^3}=0\)
CMR: \(\left(1-x^3\right)\left(1-y^3\right)\left(1-z^3\right)=\left(1-xyz\right)^3\)
Cho x,y,z>0 thỏa mãn: x+y+z=3. Tìm GTNN của \(P=\frac{\left(x+1\right)^2.\left(y+1\right)^2}{z^2+1}+\frac{\left(y+1\right)^2.\left(z+1\right)^2}{x^2+1}+\frac{\left(z+1\right)^2.\left(x+1\right)^2}{y^2+1}\)
Cho \(x,y,z\in R\)Thỏa mãn
\(\left\{{}\begin{matrix}\left(x+1\right)\left(y+1\right)\left(z+1\right)=3xyz\\\left(x^3+1\right)\left(y^3+1\right)\left(z^3+1\right)=\dfrac{81}{64}x^3y^3z^3\end{matrix}\right.\)
CMR \(xyz=0\)
Cho \(a+\dfrac{1}{b}=b+\dfrac{1}{c}=c+\dfrac{1}{a}=x\)
\(Tính\) \(P=\dfrac{2022\left(x-y\right)\left(y-z\right)\left(z-x\right)}{2xy^2+2yz^2+2zx^2+3xyz}\)
cho x y z > 0 và xyz=1. Tìm Min của \(P=\frac{x^3}{\left(1+x\right)\left(1+y\right)}+\frac{y^3}{\left(1+y\right)\left(1+z\right)}+\frac{z^3}{\left(1+z\right)\left(1+x\right)}\)
ta có : \(x^2+1=x^2+xy+yz+zx=x\left(x+y\right)+z\left(x+y\right)=\left(x+y\right)\left(x+z\right)\)
Tương tự ta đc \(y^2+1=\left(y+x\right)\left(y+z\right)\)
\(z^2+1=\left(z+x\right)\left(z+y\right)\)
ĐẶt \(A=x\sqrt{\frac{\left(1+y^2\right)\left(1+z^2\right)}{\left(1+x^2\right)}}+y\sqrt{\frac{\left(1+z^2\right)\left(1+x^2\right)}{\left(1+y^2\right)}}+z\sqrt{\frac{\left(1+x^2\right)\left(1+y^2\right)}{\left(1+z^2\right)}}\)
\(\Rightarrow A=x\sqrt{\frac{\left(x+y\right)\left(y+z\right)\left(z+x\right)\left(y+z\right)}{\left(x+y\right)\left(x+z\right)}}+y\sqrt{\frac{\left(z+x\right)\left(z+y\right)\left(x+y\right)\left(x+z\right)}{\left(x+y\right)\left(y+z\right)}}+z\sqrt{\frac{\left(x+y\right)\left(x+z\right)\left(y+z\right)\left(y+x\right)}{\left(z+x\right)\left(z+y\right)}}\)
\(\Rightarrow A=x\left(y+z\right)+y\left(x+z\right)+z\left(x+y\right)=2\left(xy+yz+zx\right)=2\)
cho 3 số thực x,y,z sao cho x+y+z=1 CMR
\(x^3+y^3+z^3-3xyz=\frac{1}{2}\left(\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\right)\)
Cho x,y,z > 0 thỏa mãn xyz = 1. Tìm GTLN:
P = \(\frac{1}{\left(3x+1\right)\left(y+z\right)+x}+\frac{1}{\left(3y+1\right)\left(z+x\right)+y}+\frac{1}{\left(3z+1\right)\left(x+y\right)+z}\)