Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Vì x>0;y>0 nên theo bất đẳng thức Cô-Si:
\(x^2+y^2\ge2\sqrt{x^2.y^2}=2xy\)
\(=>M=\frac{x^2+y^2}{xy}\ge\frac{2xy}{xy}=2\)
Dấu "=" xảy ra <=> x=y
Vậy MinM=2 khi x=y
Ta có A = 2018.2020 + 2019.2021
= (2020 - 2).2020 + 2019.(2019 + 2)
= 20202 - 2.2020 + 20192 + 2.2019
= 20202 + 20192 - 2(2020 - 2019) = 20202 + 20192 - 2 = B
=> A = B
b) Ta có B = 964 - 1= (932)2 - 12
= (932 + 1)(932 - 1) = (932 + 1)(916 + 1)(916 - 1) = (932 + 1)(916 + 1)(98 + 1)(98 - 1)
= (932 + 1)(916 + 1)(98 + 1)(94 + 1)(94 - 1)
= (932 + 1)(916 + 1)(98 + 1)(94 + 1)(92 + 1)(92 - 1)
(932 + 1)(916 + 1)(98 + 1)(94 + 1)(92 + 1).80
mà A = (932 + 1)(916 + 1)(98 + 1)(94 + 1)(92 + 1).10
=> A < B
c) Ta có A = \(\frac{x-y}{x+y}=\frac{\left(x-y\right)\left(x+y\right)}{\left(x+y\right)^2}=\frac{x^2-y^2}{x^2+2xy+y^2}< \frac{x^2-y^2}{x^2+xy+y^2}=B\)
=> A < B
d) \(A=\frac{\left(x+y\right)^3}{x^2-y^2}=\frac{\left(x+y\right)^3}{\left(x+y\right)\left(x-y\right)}=\frac{\left(x+y\right)^2}{x-y}=\frac{x^2+2xy+y^2}{x-y}< \frac{x^2-xy+y^2}{x-y}=B\)
=> A < B
\(A=\left(1-\dfrac{1}{x^2}\right)\left(1-\dfrac{1}{y^2}\right)=1+\dfrac{1}{x^2y^2}-\left(\dfrac{1}{x^2}+\dfrac{1}{y^2}\right)\)
Áp dụng bất đẳng thức Cauchy cho 2 số dương, ta có:
\(\dfrac{1}{x^2}+\dfrac{1}{y^2}\ge\dfrac{2}{xy}\) (1)
và \(x+y\ge2\sqrt{xy}\) (2)
TỪ (2) \(\Rightarrow\) \(\dfrac{1}{x^2y^2}\ge\dfrac{16}{\left(x+y\right)^4}\) và \(\dfrac{2}{xy}\ge\dfrac{8}{\left(x+y\right)^2}\)
Mặt khác, theo đề \(x+y\le1\)
=> \(\dfrac{1}{x+y}\ge1\)
=> A \(\ge1+\dfrac{16}{\left(x+y\right)^4}+\dfrac{2}{xy}\) \(\ge1+\dfrac{16}{\left(x+y\right)^4}-\dfrac{8}{\left(x+y\right)^2}\)
\(=1+16-8=9\)
Dấu ''='' xảy ra khi x = y = 0,5
Mình đánh nhầm, dòng 2 từ dưới lên phải là \(-\dfrac{2}{xy}\) nhá ! :))
By Titu's Lemma we easy have:
\(D=\left(x+\frac{1}{x}\right)^2+\left(y+\frac{1}{y}\right)^2\)
\(\ge\frac{\left(x+y+\frac{1}{x}+\frac{1}{y}\right)^2}{2}\)
\(\ge\frac{\left(x+y+\frac{4}{x+y}\right)^2}{2}\)
\(=\frac{17}{4}\)
Mk xin b2 nha!
\(P=\frac{1}{x^2+y^2}+\frac{1}{xy}+4xy=\frac{1}{x^2+y^2}+\frac{1}{2xy}+\frac{1}{2xy}+4xy\)
\(\ge\frac{\left(1+1\right)^2}{x^2+y^2+2xy}+\left(4xy+\frac{1}{4xy}\right)+\frac{1}{4xy}\)
\(\ge\frac{4}{\left(x+y\right)^2}+2\sqrt{4xy.\frac{1}{4xy}}+\frac{1}{\left(x+y\right)^2}\)
\(\ge\frac{4}{1^2}+2+\frac{1}{1^2}=4+2+1=7\)
Dấu "=" xảy ra khi: \(x=y=\frac{1}{2}\)
1/a/
\(A=\frac{2}{xy}+\frac{3}{x^2+y^2}=\left(\frac{1}{xy}+\frac{1}{xy}+\frac{4}{x^2+y^2}\right)-\frac{1}{x^2+y^2}\)
\(\ge\frac{\left(1+1+2\right)^2}{\left(x+y\right)^2}-\frac{1}{\frac{\left(x+y\right)^2}{2}}=16-2=14\)
Dấu = xảy ra khi \(x=y=\frac{1}{2}\)
b/
\(4B=\frac{4}{x^2+y^2}+\frac{8}{xy}+16xy=\left(\frac{4}{x^2+y^2}+\frac{1}{xy}+\frac{1}{xy}\right)+\left(\frac{1}{xy}+16xy\right)+\frac{5}{xy}\)
\(\ge\frac{\left(1+1+2\right)^2}{\left(x+y\right)^2}+2\sqrt{\frac{1}{xy}.16xy}+\frac{5}{\frac{\left(x+y\right)^2}{4}}\)
\(=16+8+20=44\)
\(\Rightarrow B\ge11\)
Dấu = xảy ra khi \(x=y=\frac{1}{2}\)
\(x^3 +y^3 + 3(x^2 +y^2 ) +4(x+y) + 4 = 0 \\\ \Leftrightarrow (x+y+2)[(x+1)^{2}+(y+1)^{2}-(x+1)(y+1)+1]=0\\\ \Rightarrow x+y=-2\Rightarrow \frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}=-\frac{2}{xy}\leq -\frac{2}{\frac{(x+y)^{2}}{4}}=-2\)
Dấu ''='' xảy ra khi \(x=y=-1\)
2) Viết nhầm thì phải, vế phải là 12 nhỉ
\(x\left(x-1\right)+y\left(y-1\right)=x^2+y^2-\left(x+y\right)\ge\dfrac{\left(x+y\right)^2}{2}-\left(x+y\right)\ge\dfrac{6^2}{2}-6=12\)
1) \(x\ge2y>0\Rightarrow x^3\ge8y^3\)
\(P=\dfrac{x^2+y^2}{xy}=\dfrac{x^2}{4xy}+\dfrac{x^2}{4xy}+\dfrac{x^2}{4xy}+\dfrac{x^2}{4xy}+\dfrac{4y^2}{4xy}\ge5\sqrt[5]{\dfrac{x^2}{4xy}.\dfrac{x^2}{4xy}.\dfrac{x^2}{4xy}.\dfrac{x^2}{4xy}.\dfrac{4y^2}{4xy}}=5\sqrt[5]{\dfrac{x^3}{256y^3}}\ge5\sqrt[5]{\dfrac{8y^3}{256y^3}}=5\sqrt[5]{\dfrac{1}{32}}=\dfrac{5}{2}\)