![](https://rs.olm.vn/images/avt/0.png?1311)
Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có x2+y2 / x-y = x2-2xy+y2+2xy / x-y
= (x-y)2+2xy / x-y
Mà xy = 1 => 2xy = 2. Thay vào, ta có
(x-y)2+2xy / x-y = (x-y)2+2 / x-y = (x-y)2 / x-y + 2 / x-y
= x-y + 2 / x-y
Áp dụng BĐT Cauchy, ta có
x-y + 2 / x-y ≥ 2.√(x-y).2 / x-y] = 2.√2 = (√2)3
Vậy Min A = (√2)3
![](https://rs.olm.vn/images/avt/0.png?1311)
1. \(1=x^2+y^2\ge2xy\Rightarrow xy\le\frac{1}{2}\)
\(A=-2+\frac{2}{1+xy}\ge-2+\frac{2}{1+\frac{1}{2}}=-\frac{2}{3}\)
max A = -2/3 khi x=y=\(\frac{\sqrt{2}}{2}\)
\(\frac{1}{xy}+\frac{1}{xz}=\frac{1}{x}\left(\frac{1}{y}+\frac{1}{z}\right)\ge\frac{1}{x}.\frac{4}{y+z}=\frac{4}{\left(4-t\right)t}=\frac{4}{4-\left(t-2\right)^2}\ge1\) với t = y+z => x =4 -t
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có \(x^2+y^2\ge2xy\)=>\(xy\le\frac{1}{2}\)
\(\frac{1}{A}=\frac{1}{-2xy}-\frac{1}{2}\le-1-\frac{1}{2}=-\frac{3}{2}\)
=> \(A\ge-\frac{2}{3}\)
\(MinA=-\frac{2}{3}\)khi \(x=y=\frac{\sqrt{2}}{2}\)
Trần Phúc Khang: bài này cần gì phải làm phức tạp vậy a
c/m: \(xy\le\frac{1}{2}\)( như bài Trần Phúc Khang)
Dấu "=" xảy ra <=> x=y=\(\frac{1}{\sqrt{2}}\)
\(A=\frac{-2xy}{1+xy}\ge\frac{-2.\frac{1}{2}}{1+\frac{1}{2}}=-\frac{1}{\frac{3}{2}}=-\frac{2}{3}\)
Dấu "=" xảy ra <=> x=y=\(\frac{1}{\sqrt{2}}\)
KL:.............................
![](https://rs.olm.vn/images/avt/0.png?1311)
Áp dụng bất đẳng thức AM - GM, ta được: \(2xy-4=x+y\ge2\sqrt{xy}\)
Đặt \(\sqrt{xy}=t\)thì ta có: \(2t^2-2t-4\ge0\Leftrightarrow2\left(t-2\right)\left(t+1\right)\ge0\Rightarrow t\ge2\)
\(\Rightarrow xy\ge4\)
\(P=xy+\frac{1}{x^2}+\frac{1}{y^2}\ge xy+\frac{2}{xy}=\left(\frac{2}{xy}+\frac{xy}{8}\right)+\frac{7xy}{8}\ge2\sqrt{\frac{2}{xy}.\frac{xy}{8}}+\frac{7.4}{8}=\frac{9}{2}\)
Đẳng thức xảy ra khi x = y = 2
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có: \(xy\le\frac{\left(x+y\right)^2}{4}=\frac{1}{4}\)
\(A=\frac{1}{x^2+y^2}+\frac{1}{xy}=\left(\frac{1}{x^2+y^2}+\frac{1}{2xy}\right)+\frac{1}{2xy}\)
\(\ge\frac{4}{x^2+y^2+2xy}+2=\frac{4}{\left(x+y\right)^2}+2=6\)
Dấu "=" xảy ra khio x=y=1/2
![](https://rs.olm.vn/images/avt/0.png?1311)
Điểm rơi: \(x=y=\frac{1}{2}.\)
\(A=\left(\frac{1}{x^2+y^2}+\frac{1}{2xy}\right)+\left(4xy+\frac{1}{4xy}\right)+\frac{5}{4xy}\)
\(\ge\frac{1}{x^2+y^2+2xy}+2\sqrt{4xy.\frac{1}{4xy}}+\frac{5}{\left(x+y\right)^2}\)
\(=\frac{1}{\left(x+y\right)^2}+2+\frac{5}{\left(x+y\right)^2}\ge2+\frac{6}{1^2}=8\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(P=\dfrac{1}{2\left(x^2+y^2\right)}+\dfrac{4}{xy}+2xy\)
\(\Leftrightarrow2P=\dfrac{1}{x^2+y^2}+\dfrac{8}{xy}+4xy\)
\(\Leftrightarrow2P=\dfrac{1}{x^2+y^2}+\dfrac{1}{2xy}+\dfrac{1}{4xy}+4xy+\dfrac{29}{4xy}\)
Áp dụng BĐT AM - GM , ta có :
\(\Leftrightarrow\dfrac{1}{x^2+y^2}+\dfrac{1}{2xy}+\dfrac{1}{4xy}+4xy+\dfrac{29}{4xy}\ge\dfrac{2}{\sqrt{\left(x^2+y^2\right)2xy}}+2\sqrt{\dfrac{1}{4xy}.4xy}+\dfrac{29}{4xy}\)
\(\Leftrightarrow2P\ge\)\(\dfrac{2}{\sqrt{\left(x^2+y^2\right)2xy}}+2+\dfrac{29}{4xy}\ge\dfrac{4}{\left(x+y\right)^2}+2+\dfrac{29}{\left(x+y\right)^2}\)
\(\Leftrightarrow2P\ge2+4+29=35\)
\(\Leftrightarrow P\ge\dfrac{35}{2}\)
\(\Rightarrow P_{Min}=\dfrac{35}{2}\Leftrightarrow x=y=\dfrac{1}{2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\frac{1}{x^2+y^2}+\frac{5}{xy}=\frac{1}{x^2+y^2}+\frac{1}{2xy}+\frac{9}{2xy}\ge\frac{4}{\left(x+y\right)^2}+\frac{9}{2\left(\frac{x+y}{2}\right)^2}\)
nên \(A\ge4+9.2=22\)
Dấu bằng xảy ra khi và chỉ khi \(x=y=\frac{1}{2}\)
Do x,y > 0 nên ta xét \(\frac{1}{A}=-\frac{1}{2}-\frac{1}{2xy}\)
Áp dụng bđt Cauchy ta có \(2xy\le x^2+y^2\Rightarrow\frac{1}{2xy}\ge\frac{1}{x^2+y^2}\Rightarrow-\frac{1}{2xy}\le-\frac{1}{x^2+y^2}\)
Từ đó suy ra \(\frac{1}{A}=-\frac{1}{2}-\frac{1}{2xy}\le-\frac{1}{2}-\frac{1}{x^2+y^2}=-\frac{1}{2}-1=-\frac{3}{2}\)
\(\Rightarrow A\ge-\frac{2}{3}\). Dấu "=" xảy ra khi \(x=y=\frac{\sqrt{2}}{2}\) (x,y>0)
Vậy giá trị nhỏ nhất của A bằng \(-\frac{2}{3}\) khi \(x=y=\frac{\sqrt{2}}{2}\)