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\(P=\dfrac{x^3+y^3+z^3}{xy+2yz+zx}=\dfrac{x^3}{xy+2yz+zx}+\dfrac{y^3}{xy+2yz+zx}+\dfrac{z^3}{xy+2yz+zx}\)\(\ge\sqrt[3]{\dfrac{x^3\cdot y^3\cdot z^3}{\left(xy+2yz+zx\right)^3}}=\dfrac{xyz}{xy+2yz+zx}\)
ta có: (x+y+z)^2≥0 <=>xy+yz+zx ≥\(-\dfrac{x^2+y^2+z^2}{2}\) (1)
(y+z)^2 ≥ 0 <=> yz ≥ \(-\dfrac{y^2+z^2}{2}\) (2)
(1), (2) => xy+2yz+zx ≥ \(-\dfrac{x^2}{2}\)
-.-
Mk nghĩ là x3,y3,z3.
Áp dụng BĐT AM-GM:
\(\Sigma_{cyc}\left(\frac{x^2}{\sqrt{x^3+8}}\right)=\Sigma_{cyc}\left(\frac{x^2}{\sqrt{\left(x+2\right)\left(x^2-2x+4\right)}}\right)\)\(\ge2\Sigma_{cyc}\left(\frac{x^2}{x^2-x+6}\right)\)
Áp dụng BĐT Cauchy-Schwart:
\(2\frac{\left(x+y+z\right)^2}{x^2+y^2+z^2-\left(x+y+z\right)+18}\)\(=\frac{2\left(x+y+z\right)^2}{\left(x+y+z\right)^2-2\left(xy+yz+zx\right)-\left(x+y+z\right)+18}\)\(\ge\frac{2\left(x+y+z\right)^2}{\left(x+y+z\right)^2-2\left(x+y+z\right)-\left(x+y+z\right)+18}\)
gt\(\Leftrightarrow3\left(x+y+z\right)\le3\left(xy+yz+zx\right)\le\left(x+y+z\right)^2\)
\(\Leftrightarrow\left(x+y+z\right)^2-3\left(x+y+z\right)\ge0\)
\(\Rightarrow\left[{}\begin{matrix}x+y+z\le0\\x+y+z\ge3\end{matrix}\right.\)
Đặt t=x+y+z\(\left(t\ge3\right)\)
Cần c/m:\(\frac{2t^2}{t^2-3t+18}\ge1\)
Có :\(t^2-3t+18>0\)
\(\Rightarrow2t^2\ge t^2-3t+18\)
\(\Leftrightarrow t^2+3t-18\ge3^2+3.3-18=0\)(Đúng)
Vậy min =1
Dấu = xra khi x=y=z=1.
#Walker
Kiểm tra giùm em đúng ko ạ Akai Haruma
đặt \(\left(a;b;c\right)=\left(\sqrt{\frac{yz}{x}};\sqrt{\frac{zx}{y}};\sqrt{\frac{xy}{z}}\right)\)\(\Rightarrow\)\(a^2+b^2+c^2=1\)
\(A=\Sigma\frac{1}{1-ab}=\Sigma\frac{2ab}{2\left(a^2+b^2+c^2\right)-2ab}+3\le\frac{1}{2}\Sigma\frac{\left(a+b\right)^2}{b^2+c^2+c^2+a^2}\)
\(\le\frac{1}{2}\Sigma\left(\frac{a^2}{c^2+a^2}+\frac{b^2}{b^2+c^2}\right)=\frac{9}{2}\)
dấu "=" xảy ra \(\Leftrightarrow\)\(x=y=z=\frac{1}{3}\)
Lời giải:
Áp dụng BĐT Bunhiacopxky:
\((x+y)(x+z)\geq (x+\sqrt{yz})^2\)
\(\Rightarrow \sqrt{(x+y)(y+z)(x+z)}.\frac{\sqrt{y+z}}{x}\geq \frac{(y+z)(x+\sqrt{yz})}{x}=y+z+\frac{\sqrt{yz}(y+z)}{x}\)
Hoàn toàn tương tự :
\(\sqrt{(x+y)(y+z)(x+z)}.\frac{\sqrt{x+z}}{y}\geq x+z+\frac{\sqrt{xz}(x+z)}{y}\)
\(\sqrt{(x+y)(y+z)(x+z)}.\frac{\sqrt{x+y}}{z}\geq x+y+\frac{\sqrt{xy}(x+y)}{z}\)
Cộng theo vế:
\(T\geq 2(x+y+z)+\underbrace{\frac{(x+y)\sqrt{xy}}{z}+\frac{(y+z)\sqrt{yz}}{x}+\frac{(z+x)\sqrt{zx}}{y}}_{M}\)
Ta có:
\(M=\frac{(\sqrt{2}-z)\sqrt{xy}}{z}+\frac{(\sqrt{2}-x)\sqrt{yz}}{x}+\frac{(\sqrt{2}-y)\sqrt{xz}}{y}\)
\(=\sqrt{2}\left(\frac{\sqrt{xy}}{z}+\frac{\sqrt{yz}}{x}+\frac{\sqrt{xz}}{y}\right)-(\sqrt{xy}+\sqrt{yz}+\sqrt{xz})\)
Áp dụng BĐT AM-GM:
\(\frac{\sqrt{xy}}{z}+\frac{\sqrt{yz}}{x}+\frac{\sqrt{xz}}{y}\geq 3\sqrt[3]{\frac{xyz}{xyz}}=3\)
\(\sqrt{xy}+\sqrt{yz}+\sqrt{xz}\leq \frac{x+y}{2}+\frac{y+z}{2}+\frac{z+x}{2}=x+y+z=\sqrt{2}\)
Do đó: \(M\geq 3\sqrt{2}-\sqrt{2}=2\sqrt{2}\)
\(\Rightarrow T\geq 2(x+y+z)+M\geq 2\sqrt{2}+2\sqrt{2}=4\sqrt{2}\)
Vậy \(T_{\min}=4\sqrt{2}\)
Dự đoán điểm rơi y=z=k.x
Áp dụng AM-GM:
\(2ky^2+2kz^2\ge4kyz\)
\(y^2+k^2x^2\ge2kxy\)
\(z^2+k^2x^2\ge2kxz\)
Cộng các BĐT trên theo vế:\(2k^2x^2+\left(2k+1\right)y^2+\left(2k+1\right)z^2\ge2k\left(xy+2yz+xz\right)\)
Giờ ta chỉ việc tìm k sao cho \(2k^2=2k+1\),k >0 \(\Rightarrow k=\dfrac{1+\sqrt{3}}{2}\)
\(\Rightarrow\dfrac{x^2+y^2+z^2}{xy+2yz+xz}\ge\dfrac{2k}{2k^2}=\dfrac{1}{k}=\dfrac{2}{\sqrt{3}+1}=\sqrt{3}-1\)
Dấu = xảy ra khi \(y=z=\dfrac{\sqrt{3}+1}{2}x\)
Lời giải:
Vì \(x,y,z\leq 1\Rightarrow (x-1)(y-1)(z-1)\leq 0\)
\(\Leftrightarrow (xy-x-y+1)(z-1)\leq 0\)
\(\Leftrightarrow x+y+z-xy-yz-xz+xyz-1\leq 0\)
\(\Leftrightarrow x+y+z-xy-yz-xz\leq 1-xyz\leq 1(*)\) (do \(xyz\geq 0\) )
Mặt khác:
\(y,z\in [0;1]\Rightarrow y^{2017}\leq y; z^{2018}\leq z\)
Do đó:
\(T=x+y^{2017}+z^{2018}-xy-yz-xz\leq x+y+z-xy-yz-xz(**)\)
Từ \((*);(**)\Rightarrow T\leq 1\) hay \(T_{\max}=1\)
Dấu bằng xảy ra khi \((x,y,z)=(1,1,0);(0,0,1)\) hoặc hoán vị các bộ số ấy