cho x,y,z là 3 số thực tùy ý : chứng minh \(^{x^2}\)+\(^{y^2}+z^2-yz-4x\)-3y >-7
ta có \(x^2+y^2+z^2-yz-4x-3y\)=(\(x^2-4x+4\))+(\(\dfrac{1}{4}y^2-2.\dfrac{1}{2}y.z+z^2\)) (\(\dfrac{3}{4}y^2-2.\dfrac{\sqrt{3}}{2}y.\sqrt{3}+3\))-4-3
=(x-2)\(^2\)+(\(\dfrac{1}{2}y-z\))\(^2\)+(\(\dfrac{\sqrt{3}}{2}y-\sqrt{3}\))\(^2\)-7>-7, x,y,z thuộc R
giúp vs ạ em cần gấp
cho các số thực x,y,z thoả mãn x+y+z≥6.
Tìm minP=\(\dfrac{x^2}{yz+\sqrt{1+x^3}}+\dfrac{y^2}{xz+\sqrt{1+y^3}}+\dfrac{z^2}{xy+\sqrt{1+z^3}}\)
Cho mng tham khảo ạ
Cho x, y, z > 0 thoả mãn x+y+z=2. Tìm GTNN của các biểu thức:
a) \(A=\sqrt{x^2+\dfrac{1}{x^2}}+\sqrt{y^2+\dfrac{1}{y^2}}+\sqrt{z^2+\dfrac{1}{z^2}}\)
b) \(B=\sqrt{x^2+\dfrac{1}{y^2}+\dfrac{1}{z^2}}+\sqrt{y^2+\dfrac{1}{z^2}+\dfrac{1}{x^2}}+\sqrt{z^2+\dfrac{1}{x^2}+\dfrac{1}{y^2}}\)
c) \(C=\sqrt{2x^2+\dfrac{3}{y^2}+\dfrac{4}{z}}+\sqrt{2y^2+\dfrac{3}{z^2}+\dfrac{4}{x^2}}+\sqrt{2z^2+\dfrac{3}{x^2}+\dfrac{4}{y^2}}\)
Tim x, y, z
1/ \(\sqrt{x-2}+\sqrt{y-2008}+\sqrt{z-2009}=\dfrac{1}{2}\left(x+y+z\right)\)
2/ \(x+y+z+4=2\sqrt{x-2}+4\sqrt{y-3}+6\sqrt{x-5}\)
3/ Tinh T = \(x^2+y^2+z^2-7\) biet x-y-z = \(2\sqrt{x-34}+4\sqrt{y-21}+6\sqrt{z-4}+45\)
4/ \(2x^2+9y^2-6xy-12y-6x+29=0\)
5/\(4x^2+3y-4x+4xy-10y+9=0\)
Tìm bộ ba số thực x, y, z thỏa mãn: \(\dfrac{2}{\sqrt{x}+2\sqrt{y}+3\sqrt{z}}-\dfrac{1}{2\sqrt{xy}+6\sqrt{yz}+3\sqrt{xz}}=\dfrac{1}{3}\)
Cho x, y, z > 0 thoả mãn x+y+z=1. Chứng minh rằng:
a) \(\sqrt{x^2+\dfrac{1}{x^2}}+\sqrt{y^2+\dfrac{1}{y^2}}+\sqrt{z^2+\dfrac{1}{z^2}}\ge\sqrt{82}\)
b) \(\sqrt{x^2+\dfrac{1}{x^2}+\dfrac{1}{y^2}}+\sqrt{y^2+\dfrac{1}{y^2}+\dfrac{1}{z^2}}+\sqrt{z^2+\dfrac{1}{z^2}+\dfrac{1}{x^2}}\ge\sqrt{163}\)
c)\(\sqrt{x^2+\dfrac{2}{y^2}+\dfrac{3}{z^2}}+\sqrt{y^2+\dfrac{2}{z^2}+\dfrac{3}{x^2}}+\sqrt{z^2+\dfrac{2}{z^2}+\dfrac{3}{y^2}}\ge\sqrt{406}\)
1) \(\dfrac{x-3x^2}{2}+\sqrt{2x^4-x^3+7x^2-3x+3}=2\)
2) \(1+\sqrt{\dfrac{x-2}{1-x}}=\dfrac{2x^2-2x+1}{x^2-2x+2}\)
3) \(x+y+z+\dfrac{3}{x-1}+\dfrac{3}{y-1}+\dfrac{3}{z-1}=2\left(\sqrt{x+2}+\sqrt{y+2}+\sqrt{z+2}\right)\) với x ,y ,z > 1
4) \(\sqrt[3]{x+6}+x^2=7-\sqrt{x-1}\)
5) \(x^4-2x^3+x-\sqrt{2\left(x^2-x\right)}=0\)
Cho xy + yz + xz = 1. Chứng minh:
\(\dfrac{x}{\sqrt{x^2+1}}+\dfrac{y}{\sqrt{y^2+1}}+\dfrac{z}{\sqrt{z^2+1}}\le\dfrac{3}{2}\)
Cho x\(\ge\)3; y\(\ge2\); z\(\ge\)1. Chứng minh rằng:
\(\dfrac{xy\sqrt{x-1}+zx\sqrt{y-2}+yz\sqrt{z-3}}{xyz}\le\dfrac{1}{2}+\dfrac{\sqrt{2}}{4}+\dfrac{\sqrt{3}}{6}\)
Cho x, y, z >0 thỏa mãn x + y + z= xyz
CMR: \(\dfrac{x}{x^2+yz}+\dfrac{y}{y^2+xz}+\dfrac{z}{z^2+xy}\le\dfrac{\sqrt{3}}{2}\)