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Dự đoán dấu = xảy ra khi x=y=\(\dfrac{z}{2}\)
ta có: \(VT=3+\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}+\dfrac{y^2}{z^2}+\dfrac{z^2}{y^2}+\dfrac{x^2}{z^2}+\dfrac{z^2}{x^2}\)
\(=3+\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right)+\left(\dfrac{y^2}{z^2}+\dfrac{x^2}{z^2}\right)+\left(\dfrac{z^2}{y^2}+\dfrac{z^2}{x^2}\right)\)
Áp dụng BĐT AM-GM: \(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\ge2\)
Áp dụng BĐT bunyakovsky:\(\dfrac{y^2}{z^2}+\dfrac{x^2}{z^2}\ge\dfrac{1}{2}\left(\dfrac{y}{z}+\dfrac{x}{z}\right)^2=\dfrac{1}{2}.\dfrac{\left(x+y\right)^2}{z^2}\)
\(\dfrac{z^2}{x^2}+\dfrac{z^2}{y^2}\ge\dfrac{1}{2}\left(\dfrac{z}{x}+\dfrac{z}{y}\right)^2\ge\dfrac{1}{2}\left(\dfrac{4z}{x+y}\right)^2=\dfrac{8z^2}{\left(x+y\right)^2}\)(AM-GM)
do đó \(VT\ge5+\dfrac{1}{2}\dfrac{\left(x+y\right)^2}{z^2}+\dfrac{8z^2}{\left(x+y\right)^2}\)
Đặt \(\dfrac{z}{x+y}=a\)(a>0)thì \(a\ge1\)do \(z\ge x+y\)
\(VT\ge8a^2+\dfrac{1}{2a^2}+5=\dfrac{a^2}{2}+\dfrac{1}{2a^2}+\dfrac{15}{2}a^2+5\ge\dfrac{a^2}{2}+\dfrac{1}{2a^2}+\dfrac{25}{2}\)
Áp dụng BĐT AM-GM: \(\dfrac{a^2}{2}+\dfrac{1}{2a^2}\ge2\sqrt{\dfrac{a^2}{4a^2}}=1\)
do đó \(VT\ge1+\dfrac{25}{2}=\dfrac{27}{2}\)(đpcm)
Dấu = xảy ra khi a=1 hay \(x=y=\dfrac{z}{2}\)
Ta có:
\(\dfrac{x^2}{\sqrt{1-x^2}}=\dfrac{x^3}{x\sqrt{1-x^2}}\)
Áp dụng BĐT Cosi ta có:
\(x\sqrt{1-x^2}\le\dfrac{x^2+1-x^2}{2}=\dfrac{1}{2}\)
\(\Rightarrow\dfrac{x^3}{x\sqrt{1-x^2}}\ge2x^3\)
Cmtt:
\(\dfrac{y^3}{y\sqrt{1-y^2}}\ge2y^3\)
\(\dfrac{z^3}{z\sqrt{1-z^2}}\ge2z^3\)
\(\Rightarrow\dfrac{x^2}{\sqrt{1-x^2}}+\dfrac{y^2}{\sqrt{1-y^2}}+\dfrac{z^2}{\sqrt{1-z^2}}=\dfrac{x^3}{x\sqrt{1-x^2}}+\dfrac{y^3}{y\sqrt{1-y^2}}+\dfrac{z^3}{z\sqrt{1-z^2}}\ge2\left(x^3+y^3+z^3\right)=2\) (ĐPCM)
\(P=1+\dfrac{z}{x}+\dfrac{z}{y}+\dfrac{z^2}{xy}\)
vì \(x^2+y^2=z^2\Rightarrow z=\sqrt{x^2+y^2}\)
Áp dụng BĐT bunyakovsky:
\(\left(1+1\right)\left(x^2+y^2\right)\ge\left(x+y\right)^2\)
\(\Rightarrow z=\sqrt{x^2+y^2}\ge\sqrt{\dfrac{\left(x+y\right)^2}{2}}=\dfrac{\sqrt{2}\left(x+y\right)}{2}\)
do đó \(P\ge1+\dfrac{\sqrt{2}\left(x+y\right)}{2}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)+\dfrac{x^2+y^2}{xy}\)
Áp dụng BĐT cauchy:\(\dfrac{1}{x}+\dfrac{1}{y}\ge\dfrac{4}{x+y}\)và\(x^2+y^2\ge2xy\)
\(P\ge1+\dfrac{\sqrt{2}\left(x+y\right)}{2}.\dfrac{4}{x+y}+\dfrac{2xy}{xy}=3+2\sqrt{2}\)
dấu = xảy ra khi \(x=y=\dfrac{\sqrt{2}z}{2}\)
Hình như đề có vấn đề đó bạn
theo mình
Có : x+y+z =1
\(\Rightarrow\)\(x^2+y^2+z^2+2xz+2yz+2xy=1\)
\(\Leftrightarrow\)xy+xz+zy =0
Lại có : \(x^3+y^3+z^3-3xyz=\left(x+y\right)^3+z^3-3xy\left(x+y+z\right)=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)=1\left(1-0\right)=1\)
\(x^3+y^3+z^3=1+3=4\)
\(\Rightarrow\dfrac{1}{x^3}+\dfrac{1}{y^3}+\dfrac{1}{z^3}=4\)
\(\dfrac{1}{x^3}+\dfrac{1}{y^3}+\dfrac{1}{z^3}=\dfrac{\left(yz\right)^3+\left(xz\right)^3+\left(xy\right)^3}{x^3y^3z^3}=\left(yz\right)^3+\left(xz\right)^3+\left(xy\right)^3\)
\(=\left(xy+yz+zx\right)\left[\left(xy\right)^2+\left(yz\right)^2+\left(zx\right)^2-xy^2z-xyz^2-x^2yz\right]+3xy.yz.zx\)
\(=0+3=3\)
\(\frac{1}{x^2+y^2}+\frac{1}{y^2+z^2}+\frac{1}{z^2+x^2}=\frac{x^2+y^2+z^2}{x^2+y^2}+\frac{x^2+y^2+z^2}{y^2+z^2}+\frac{x^2+y^2+z^2}{z^2+x^2}\)
\(=1+\frac{z^2}{x^2+y^2}+1+\frac{x^2}{y^2+z^2}+1+\frac{y^2}{z^2+x^2}\)
\(\le3+\frac{z^2}{2xy}+\frac{x^2}{2yz}+\frac{y^2}{2zx}\)\(=3+\frac{x^3+y^3+z^3}{2xyz}\)
Dấu "=" \(\Leftrightarrow x=y=z=\frac{\sqrt{3}}{3}\)
ta có: \(VT=\frac{x^2+y^2+z^2}{x^2+y^2}+\frac{x^2+y^2+z^2}{y^2+z^2}+\frac{x^2+y^2+z^2}{z^2+x^2}=3+\frac{z^2}{x^2+y^2}+\frac{x^2}{y^2+z^2}+\frac{y^2}{x^2+z^2}\)
Áp dụng bất đẳng thức cauchy: \(\hept{\begin{cases}x^2+y^2\ge2xy\\y^2+z^2\ge2yz\\z^2+x^2\ge2xz\end{cases}}\)
do đó \(VT\le3+\frac{x^2}{2yz}+\frac{y^2}{2xz}+\frac{z^2}{2xy}=\frac{x^3+y^3+z^3}{2xyz}+3=VF\)
đẳng thức xảy ra khi x=y=z
đề đúng chứ bạn
thiếu, đẳng thức xảy ra khi nào?