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đặt \(a=x^2,b=y^2\left(a,b\ge0\right)\)thì \(P=\frac{\left(a-b\right)\left(1-ab\right)}{\left(1+a\right)^2\left(1+b\right)^2}\)
Zì \(a,b\ge0\)nên
\(\left(a-b\right)\left(1-ab\right)=a-a^2b-b+ab^2\le a+ab^2=a\left(1+b^2\right)\le a\left(1+2b+b^2\right)=a\left(1+b\right)^2\)
Lại có \(\left(1+a\right)^2=\left(1-a\right)^2+4a\ge4a\)
=>\(P\le\frac{a\left(1+b\right)^2}{4a\left(1+b\right)^2}=\frac{1}{4}\)
dấu "=" xảy ra khi zà chỉ khi\(\hept{\begin{cases}a=1\\b=0\end{cases}=>\hept{\begin{cases}x=\pm1\\y=0\end{cases}}}\)
zậy \(maxP=\frac{1}{4}khi\hept{\begin{cases}x=\pm1\\y=0\end{cases}}\)
Áp dụng BĐT AM-GM: \(\left(x^2-y^2\right)\left(1-x^2y^2\right)\le\frac{1}{4}\left(x^2-y^2+1-x^2y^2\right)^2=\frac{1}{4}\left(1-y^2\right)^2\left(1+x^2\right)^2\)
\(P\le\frac{1}{4}\frac{\left(1-y^2\right)^2}{\left(1+y^2\right)^2}\)
mà theo BĐT AM-GM:\(\left(1-y\right)\left(1+y\right)\le\frac{1}{4}\left(1-y+1+y\right)^2=1\)
\(\Rightarrow P\le\frac{1}{4}.\frac{1}{\left(1+y^2\right)^2}\le\frac{1}{4}.\frac{1}{1}=\frac{1}{4}\)
Dấu = xảy ra khi x=1;y=0 wait : có gì đó sai sai. số thực
Ta có BĐT:
\(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}\le\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\)
\(\Leftrightarrow6\left(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}\right)+2016\le6\left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\right)+2016\)
\(\Leftrightarrow7.\left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\right)\le6\left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\right)+2016\)
\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\le2016\)
Xét \(P=\frac{1}{\sqrt{3\left(2x^2+y^2\right)}}+\frac{1}{\sqrt{3\left(2y^2+z^2\right)}}+\frac{1}{\sqrt{3\left(2z^2+x^2\right)}}\)
\(P^2=\left(\frac{1}{\sqrt{3}}.\frac{1}{\sqrt{2x^2+y^2}}+\frac{1}{\sqrt{3}}.\frac{1}{\sqrt{2y^2+z^2}}+\frac{1}{\sqrt{3}}.\frac{1}{\sqrt{2z^2+x^2}}\right)^2\)
Áp dụng BĐT Bunhiacopxki ta có:
\(P^2\le\left(\left(\frac{1}{\sqrt{3}}\right)^2+\left(\frac{1}{\sqrt{3}}\right)^2+\left(\frac{1}{\sqrt{3}}\right)^2\right)\left(\left(\frac{1}{\sqrt{2x^2+y^2}}\right)^2+\left(\frac{1}{\sqrt{2y^2+z^2}}\right)^2+\left(\frac{1}{\sqrt{2z^2+x^2}}\right)^2\right)\)
\(\Leftrightarrow P^2\le\frac{1}{2x^2+y^2}+\frac{1}{2y^2+z^2}+\frac{1}{2z^2+x^2}\)
Mặt khác ta có:
\(\frac{1}{2x^2+y^2}=\frac{1}{x^2+x^2+y^2}\le\frac{1}{9}\left(\frac{1}{x^2}+\frac{1}{x^2}+\frac{1}{y^2}\right)\)
\(\frac{1}{2y^2+z^2}\le\frac{1}{9}\left(\frac{1}{y^2}+\frac{1}{y^2}+\frac{1}{z^2}\right)\)
\(\frac{1}{2z^2+x^2}\le\frac{1}{9}\left(\frac{1}{z^2}+\frac{1}{z^2}+\frac{1}{x^2}\right)\)
\(\Rightarrow P^2\le\frac{1}{3}\left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\right)\le\frac{1}{3}.2016=672\)
\(\Rightarrow P\le4\sqrt{42}\)
Dấu '=' xảy ra khi \(x=y=z=\sqrt{\frac{1}{672}}\)
\(\left(x^2;y^2\right)=\left(a;b\right)\Rightarrow P=\dfrac{\left(a-b\right)\left(1-ab\right)}{\left(1+a\right)^2\left(1+b\right)^2}\)
Ta có:
\(\left(a+b\right)\left(1+ab\right)-\left(a-b\right)\left(1-ab\right)=2b\left(a^2+1\right)\ge0;\forall a;b\ge0\)
\(\Rightarrow\left(a+b\right)\left(1+ab\right)\ge\left(a-b\right)\left(1-ab\right)\)
\(\Rightarrow P\le\dfrac{\left(a+b\right)\left(1+ab\right)}{\left(1+a\right)^2\left(1+b\right)^2}\le\dfrac{\left(a+b+1+ab\right)^2}{4\left(1+a\right)^2\left(1+b\right)^2}=\dfrac{1}{4}\)
\(P_{max}=\dfrac{1}{4}\) khi \(\left(a;b\right)=\left(1;0\right)\) hay \(\left(x;y\right)=\left(1;0\right)\)
\(P=\dfrac{\left[\left(x-y\right)\left(1+xy\right)\right]\left[\left(x+y\right)\left(1-xy\right)\right]}{\left(1+x^2\right)^2\left(1+y^2\right)^2}\)
Áp dụng BĐT Cosi ta có:
\(\left(x-y\right)\left(1+xy\right)\le\dfrac{\left(x-y\right)^2+\left(1+xy\right)^2}{2}=\dfrac{\left(1+x^2\right)\left(1+y^2\right)}{2}\\ \left(x+y\right)\left(1-xy\right)\le\dfrac{\left(x+y\right)^2+\left(1-xy\right)^2}{2}=\dfrac{\left(1+x^2\right)\left(1+y^2\right)}{2}\)
\(\to P\le\dfrac{\left(1+x^2\right)^2\left(1+y^2\right)^2}{4\left(1+x^2\right)^2\left(1+y^2\right)^2}=\dfrac{1}{4}\)
Dấu \("="\Leftrightarrow\left(x;y\right)=\left(1;0\right)\)