Các câu hỏi tương tự
cho x,y,z dương thỏa mãn \(xy+yz+zx=3\)Tìm min \(P=\frac{x^2}{\sqrt{x^3+8}}+\frac{y^2}{\sqrt{y^3+8}}+\frac{z^2}{\sqrt{z^3+8}}\)
Cho các số thực dương x,y,z thỏa mãn xy+yz+zx>=x+y+z
Chứng minh rằng \(\frac{x^2}{\sqrt{x^3+8}}+\frac{y^2}{\sqrt{y^3+8}}+\frac{z^2}{\sqrt{z^3+8}}\ge1\)
Cho x,y,z>0 thỏa mãn xy+yz+zx=3
Tìm GTLN của C=\(\frac{x^2}{\sqrt{x^3+8}}\)+\(\frac{y^2}{\sqrt{y^3+8}}\)+\(\frac{z^2}{\sqrt{z^3+8}}\)
đặt Afrac{sqrt{yz}}{x+3sqrt{yz}}+frac{sqrt{zx}}{y+3sqrt{zx}}+frac{sqrt{xy}}{z+3sqrt{xy}}Rightarrow1-3Afrac{x}{x+3sqrt{yz}}+frac{y}{y+3sqrt{zx}}+frac{z}{z+3sqrt{xy}}gefrac{x}{x+frac{3}{2}left(y+zright)}+frac{y}{y+frac{3}{2}left(z+xright)}+frac{z}{z+frac{3}{2}left(x+yright)}frac{2x}{2x+3left(y+zright)}+frac{2y}{2y+3left(z+xright)}+frac{2z}{2z+3left(x+yright)}frac{2x^2}{2x^2+3xy+3xz}+frac{2y^2}{2y^2+3yz+3xy}+frac{2z^2}{2z^2+3zx+3yz}gefrac{2left(x+y+zright)^2}{2left(x^2+y^2+z^2right)+6left(xy+yz+zxr...
Đọc tiếp
đặt \(A=\frac{\sqrt{yz}}{x+3\sqrt{yz}}+\frac{\sqrt{zx}}{y+3\sqrt{zx}}+\frac{\sqrt{xy}}{z+3\sqrt{xy}}\)
\(\Rightarrow1-3A=\frac{x}{x+3\sqrt{yz}}+\frac{y}{y+3\sqrt{zx}}+\frac{z}{z+3\sqrt{xy}}\)
\(\ge\frac{x}{x+\frac{3}{2}\left(y+z\right)}+\frac{y}{y+\frac{3}{2}\left(z+x\right)}+\frac{z}{z+\frac{3}{2}\left(x+y\right)}\)
\(=\frac{2x}{2x+3\left(y+z\right)}+\frac{2y}{2y+3\left(z+x\right)}+\frac{2z}{2z+3\left(x+y\right)}\)
\(=\frac{2x^2}{2x^2+3xy+3xz}+\frac{2y^2}{2y^2+3yz+3xy}+\frac{2z^2}{2z^2+3zx+3yz}\)
\(\ge\frac{2\left(x+y+z\right)^2}{2\left(x^2+y^2+z^2\right)+6\left(xy+yz+zx\right)}=\frac{2\left(x+y+z\right)^2}{2\left(x+y+z\right)^2+2\left(xy+yz+zx\right)}\)
\(\ge\frac{2\left(x+y+z\right)^2}{2\left(x+y+z\right)^2+\frac{2}{3}\left(x+y+z\right)^2}=\frac{2\left(x+y+z\right)^2}{\frac{8}{3}\left(x+y+z\right)^2}=\frac{3}{4}\)
\(\Rightarrow1-3A\ge\frac{3}{4}\Rightarrow A\le\frac{3}{4}\left(Q.E.D\right)\)
12 tháng 12 2018 lúc 21:56
cho x;y;z là 3 số thực dương
Tìm min \(S=\frac{\sqrt{x^2-xy+y^2}}{x+y+2z}+\frac{\sqrt{y^2-yz+z^2}}{y+z+2x}+\frac{\sqrt{z^2-zx+x^2}}{z+x+2y}\)
Help me~
x,y,z, dương tm:x+y+z>=3. Tìm GTNN của P= \(\frac{x^2}{yz+\sqrt{8+x^3}}+\frac{y^2}{xz+\sqrt{8+y^3}}+\frac{z^2}{xy+\sqrt{8+z^3}}\)
cho x+y+z=\(\frac{3}{2}\)TÌm min
P=\(\frac{\sqrt{x^2+xy+y^2}}{4yz+1}+\frac{\sqrt{y^2+yz+z^2}}{4zx+1}+\frac{\sqrt{z^2+zx+x^2}}{4xy+1}\)
Cho x,y,z>0 :\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\sqrt{3}\)
Tìm min P=\(\frac{\sqrt{2x^2+y^2}}{xy}+\frac{\sqrt{2y^2+z^2}}{yz}+\frac{\sqrt{2z^2+x^2}}{zx}\)
Cho x,y,z > 0 ; x + y + z = 1
CMR: \(\sqrt{\frac{xy}{z+xy}}+\sqrt{\frac{yz}{x+yz}}+\sqrt{\frac{zx}{y+zx}}\le\frac{3}{2}\)