Cho x, y, z là các số thực dương thỏa mãn \(x\ge z\). CMR:
\(\dfrac{xz}{y^2+yz}+\dfrac{y^2}{xz+yz}+\dfrac{x+2z}{x+z}\ge\dfrac{5}{2}\)
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29 tháng 4 2020
Đặt \(H=\frac{xz}{y^2+yz}+\frac{y^2}{zx+yz}+\frac{x+2z}{x+z}\)
\(=\frac{1}{\frac{y^2}{xz}+\frac{yz}{xz}}+\frac{1}{\frac{zx}{y^2}+\frac{yz}{y^2}}+\frac{x+z+z}{x+z}\)
\(=\frac{1}{\frac{y^2}{zx}+\frac{y}{x}}+\frac{1}{\frac{zx}{y^2}+\frac{z}{y}}+\frac{1}{\frac{x}{z}+1}+1\)
Đặt \(\frac{x}{y}=a;\frac{y}{z}=b\Rightarrow ab=\frac{x}{z}\ge1\)
Khi đó \(H=\frac{1}{\frac{b}{a}+\frac{1}{a}}+\frac{1}{\frac{a}{b}+\frac{1}{b}}+\frac{1}{ab+1}+1\)
\(=\frac{a}{b+1}+\frac{b}{a+b}+\frac{1}{ab+1}+1\)
Ta cần chứng minh \(U=\frac{a}{b+c}+\frac{b}{a+b}+\frac{1}{ab+1}\ge\frac{3}{2}\)
\(\Leftrightarrow\left(\frac{a}{b+1}+1\right)+\left(\frac{b}{a+1}+1\right)+\frac{1}{ab+1}\ge\frac{7}{2}\)
\(\Leftrightarrow\frac{a+b+1}{b+1}+\frac{a+b+1}{a+1}+\frac{1}{ab+1}\ge\frac{7}{2}\)
\(\Leftrightarrow\left(a+b+1\right)\left(\frac{1}{b+1}+\frac{1}{a+1}\right)+\frac{1}{ab+1}\ge\frac{7}{2}\)
Khi đó \(Y=\left(a+b+1\right)\left(\frac{1}{a+1}+\frac{1}{b+1}\right)+\frac{1}{ab+1}\)
\(\ge\left(a+b+1\right)\cdot\frac{4}{a+b+2}+\frac{1}{ab+1}\)
\(\ge\frac{4\left(a+b+1\right)}{a+b+2}+\frac{1}{\frac{\left(a+b\right)^2}{4}+1}\)
Đặt \(t=a+b\ge2\sqrt{ab}\ge2\)
Ta cần chứng minh \(\frac{4\left(t+1\right)}{t+2}+\frac{1}{\frac{t^2}{4}+1}\ge\frac{7}{2}\)
\(\Leftrightarrow\frac{\left(t-2\right)^3}{2\left(t+2\right)\left(t^2+4\right)}\ge0\) ( đúng )
Vậy ta có đpcm.
29 tháng 4 2020
ta có:
\(\frac{xz}{y^2+yz}+\frac{y^2}{xz+yz}+\frac{z+2z}{z+x}=\frac{\frac{xz}{yz}}{\frac{y^2}{yz}+1}+\frac{\frac{y^2}{yz}}{\frac{xz}{yz}+1}+\frac{1+\frac{2z}{x}}{1+\frac{z}{x}}\)\(=\frac{\frac{x}{y}}{\frac{y}{z}+1}+\frac{\frac{y}{z}}{\frac{x}{y}+1}+\frac{1+\frac{2z}{x}}{1+\frac{z}{x}}=\frac{a^2}{b^2+1}+\frac{b^2}{a^2+1}+\frac{1+2c^2}{1+c^2}\)
trong đó \(a^2=\frac{x}{y};b^2=\frac{y}{z};c^2=\frac{z}{x}\left(a;b;c>0\right)\)
Nhận xét rằng \(a^2\cdot b^2=\frac{x}{z}=\frac{1}{c^2}\ge1\)(do x>=z)
Xét \(\frac{a^2}{b^2+1}+\frac{b^2}{a^2+1}+\frac{c^2}{ab+1}\)\(=\frac{a^2\left(a^2+1\right)\left(ab+1\right)+b^2\left(b^2+1\right)\left(ab+1\right)-2aba^2\left(a^2+1\right)\left(b^2+1\right)}{\left(a^2+1\right)\left(b^2+1\right)\left(ab+1\right)}\)
\(=\frac{ab\left(a^2-b^2\right)+\left(a-b\right)\left(a^3-b^3\right)+\left(a-b\right)^2}{\left(a^2+1\right)\left(b^2+1\right)\left(ab+1\right)}\ge0\)
Do đó: \(\frac{a^2}{b^2+1}+\frac{b^2}{a^2+1}\ge\frac{2ab}{ab+1}=\frac{\frac{2}{c}}{\frac{1}{c}+1}=\frac{2}{1+c}\left(1\right)\)đẳng thức xảy ra <=> a=b
khi đó:
\(\frac{2}{1+c}+\frac{1+2c^2}{c^2+1}-\frac{5}{2}=\frac{2\left[2\left(1+c^2\right)+\left(1+c\right)\left(1+2c^2\right)\right]-5\left(1+c\right)\left(1+c^2\right)}{2\left(1+c\right)\left(1+c^2\right)}\)
\(=\frac{1-3c+3c^2-c^3}{2\left(1+c\right)\left(1+c^2\right)}=\frac{\left(1-c\right)^3}{2\left(1+c\right)\left(1+c^2\right)}\ge0\)(do c=<1) (2)
Từ (1) và (2) => đpcm
Đẳng thức xảy ra <=> a=b, c=1 <=> x=y=z
DT
9 tháng 1 2018
Bài này cũng dễ mà:
Áp dụng BĐT Cô-si, ta có:
\(y+z+1\ge3\sqrt[3]{yz}\)
\(\Rightarrow\)\(\dfrac{y+z+1}{3}\ge\sqrt[3]{yz}\)
\(\Rightarrow\)\(\dfrac{x}{\sqrt[3]{yz}}\ge\dfrac{3x}{y+z+1}\)
\(\Rightarrow\)\(\sum\dfrac{x}{\sqrt[3]{yz}}\ge\sum\dfrac{3x}{y+z+1}\)
Mà \(\sum\dfrac{3x}{y+z+1}=\sum\dfrac{3x^2}{xy+xz+x}\)
Áp dụng BĐT Cauchy -Schwaz:
\(\sum\dfrac{3x^2}{xy+xz+x}\ge\dfrac{3\left(x+y+z\right)^2}{2\left(xy+yz+xz\right)+x+y+z}\)
Mà:
\(xy+yz+xz\le x^2+y^2+z^2\)(BĐT phụ)
\(\Rightarrow\)\(2\left(xy+yz+xz\right)\le2\left(x^2+y^2+z^2\right)=6\)
Áp dụng BĐT Bunhicopski:
\(\left(x+y+z\right)^2\le3\left(x^2+y^2+z^2\right)=9\)
\(\Rightarrow x+y+z\le3\)
\(\Rightarrow2\left(xy+yz+xz\right)+x+y+z\le6+3=9\)
\(\Rightarrow\)\(\dfrac{3\left(x+y+z\right)^2}{2\left(xy+yz+xz\right)+x+y+z}\ge\dfrac{3\left(x+y+z\right)^2}{9}\ge\dfrac{\left(x+y+z\right)^2}{3}\ge xy+yz+xz\left(ĐPCM\right)\)
Dấu "=" xảy ra \(\Leftrightarrow\)x=y=z=1
HN
26 tháng 8 2017
Đặt cái ban đầu là P
Ta có: \(xy+yz+zx=xyz\)
\(\Leftrightarrow\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1\)
Ta lại có:
\(\dfrac{xy}{z^3\left(1+x\right)\left(1+y\right)}+\dfrac{1+x}{64x}+\dfrac{1+y}{64y}\ge\dfrac{3}{16z}\)
\(\Leftrightarrow\dfrac{xy}{z^3\left(1+x\right)\left(1+y\right)}\ge\dfrac{3}{16z}-\dfrac{1}{32}-\dfrac{1}{64x}-\dfrac{1}{64y}\left(1\right)\)
Tương tự ta có:
\(\left\{{}\begin{matrix}\dfrac{yz}{x^3\left(1+y\right)\left(1+z\right)}\ge\dfrac{3}{16x}-\dfrac{1}{32}-\dfrac{1}{64y}-\dfrac{1}{64z}\left(2\right)\\\dfrac{zx}{y^3\left(1+z\right)\left(1+x\right)}\ge\dfrac{3}{16y}-\dfrac{1}{32}-\dfrac{1}{64z}-\dfrac{1}{64x}\left(3\right)\end{matrix}\right.\)
Từ (1), (2), (3) ta có:
\(P\ge\dfrac{3}{16}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)-\dfrac{1}{32}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)-\dfrac{3}{32}\)
\(=\dfrac{3}{16}-\dfrac{1}{32}-\dfrac{3}{32}=\dfrac{1}{16}\)
Dấu = xảy ra khi \(x=y=z=3\)
thật vĩ đại Hung nguyen
PL
0
N
12 tháng 3 2017
solution:
ta có: \(3=x^2+y^2+z^2\ge3\sqrt[3]{x^2y^2z^2}\Leftrightarrow xyz\le1\)(theo BĐT cauchy cho 3 số )
\(\Rightarrow xy\le\dfrac{1}{z};yz\le\dfrac{1}{x};xz\le\dfrac{1}{y}\)
\(\Rightarrow\dfrac{x}{\sqrt[3]{yz}}\ge\dfrac{x}{\dfrac{1}{\sqrt[3]{x}}}=x\sqrt[3]{x}=\sqrt[3]{x^4}\)
tương tự ta có:\(\dfrac{y}{\sqrt[3]{xz}}\ge\sqrt[3]{y^4};\dfrac{z}{\sqrt[3]{xy}}\ge\sqrt[3]{z^4}\)
cả 2 vế các BĐT đều dương,cộng vế với vế:
\(S=\dfrac{x}{\sqrt[3]{yz}}+\dfrac{y}{\sqrt[3]{xz}}+\dfrac{z}{\sqrt[3]{xy}}\ge\sqrt[3]{x^4}+\sqrt[3]{y^4}+\sqrt[3]{z^4}\)
Áp dụng BĐT bunyakovsky ta có:
\(\left(\sqrt[3]{x^4}+\sqrt[3]{y^4}+\sqrt[3]{z^4}\right)\left(x^2+y^2+z^2\right)\ge\left(\sqrt[3]{x^8}+\sqrt[3]{y^8}+\sqrt[3]{z^8}\right)^2=\left(x^2+y^2+z^2\right)^2\)
\(\Rightarrow S\ge x^2+y^2+z^2\)
đến đây ta lại có BĐT quen thuộc: \(x^2+y^2+z^2\ge xy+yz+xz\)
\(\Rightarrow S\ge xy+yz+xz\left(đpcm\right)\)
dấu = xảy ra khi và chỉ khi x=y=z mà x2+y2+z2=3 => x=y=z=1
*cách khác : Áp dụng BĐT cauchy - schwarz(bunyakovsky):
\(S=\dfrac{x}{\sqrt[3]{yz}}+\dfrac{y}{\sqrt[3]{xz}}+\dfrac{z}{\sqrt[3]{xy}}=\dfrac{x^4}{x^3.\dfrac{1}{\sqrt[3]{x}}}+\dfrac{y^4}{y^3.\dfrac{1}{\sqrt[3]{y}}}+\dfrac{z^4}{z^3.\dfrac{1}{\sqrt[3]{z}}}\)
\(S\ge\dfrac{\left(x^2+y^2+z^2\right)^2}{x^2+y^2+z^2}=x^2+y^2+z^2\ge xy+yz+xz\)
AT
14 tháng 7 2021
\(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\ge\dfrac{1}{\sqrt{xy}}+\dfrac{1}{\sqrt{yz}}+\dfrac{1}{\sqrt{zx}}\)
\(\Rightarrow\dfrac{2}{x}+\dfrac{2}{y}+\dfrac{2}{z}\ge\dfrac{2}{\sqrt{xy}}+\dfrac{2}{\sqrt{yz}}+\dfrac{2}{\sqrt{zx}}\)
\(\Rightarrow\dfrac{2}{x}+\dfrac{2}{y}+\dfrac{2}{z}-\dfrac{2}{\sqrt{xy}}+\dfrac{2}{\sqrt{yz}}+\dfrac{2}{\sqrt{zx}}\ge0\)
\(\Rightarrow\dfrac{1}{x}-\dfrac{2}{\sqrt{xy}}+\dfrac{1}{y}+\dfrac{1}{y}-\dfrac{2}{\sqrt{yz}}+\dfrac{1}{z}+\dfrac{1}{z}-\dfrac{2}{\sqrt{zx}}+\dfrac{1}{x}\ge0\)
\(\Rightarrow\left(\dfrac{1}{\sqrt{x}}-\dfrac{1}{\sqrt{y}}\right)^2+\left(\dfrac{1}{\sqrt{y}}-\dfrac{1}{\sqrt{z}}\right)^2+\left(\dfrac{1}{\sqrt{z}}-\dfrac{1}{\sqrt{x}}\right)^2\ge0\) (luôn đúng)
Dấu = xảy ra khi \(x=y=z\)
8 tháng 12 2023
Có \(VT=\dfrac{x^2}{x^3-xyz+2013x}+\dfrac{y^2}{y^3-xyz+2013y}+\dfrac{z^2}{z^3-xyz+2013z}\)
\(\ge\dfrac{\left(x+y+z\right)^2}{x^3+y^3+z^3-3xyz+2013\left(x+y+z\right)}\)
\(=\dfrac{\left(x+y+z\right)^2}{\left(x+y+z\right)\left[x^2+y^2+z^2-\left(xy+yz+zx\right)\right]+2013\left(x+y+z\right)}\)
\(=\dfrac{x+y+z}{x^2+y^2+z^2-\left(xy+yz+zx\right)+3\left(xy+yz+zx\right)}\)
(vì \(2013=3.671=3\left(xy+yz+zx\right)\))
\(=\dfrac{x+y+z}{x^2+y^2+z^2+2\left(xy+yz+zx\right)}\)
\(=\dfrac{x+y+z}{\left(x+y+z\right)^2}\)
\(=\dfrac{1}{x+y+z}\)
ĐTXR \(\Leftrightarrow\dfrac{1}{x^2-yz+2013}=\dfrac{1}{y^2-zx+2013}=\dfrac{1}{z^2-xy+2013}\)
\(\Leftrightarrow x^2-yz=y^2-zx=z^2-xy\)
\(\Leftrightarrow x=y=z\) (với \(x,y,z>0\))
Vậy ta có đpcm.
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