Cho A = (x+1).(y+1),trong đó: x.y = 1 (x>0, y>0). Chứng minh rằng: A \(\ge\) 4
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Ta có:
A = (x + 1)(y + 1)
=> A = xy + x + y +1
=> A = 1 + x + y + 1
=> A = 2 + x + y
Vì x > 0 ; y > 0
=>x \(\ge\)1; y\(\ge\)1
=> x + y \(\ge\)2
=> 2 + x + y \(\ge\)4
hay A \(\ge\)4
![](https://rs.olm.vn/images/avt/0.png?1311)
1.Ta có :\(x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)\)
\(=x^2-xy+y^2\) (do x+y=1)
\(=\dfrac{3}{4}\left(x-y\right)^2+\dfrac{1}{4}\left(x+y\right)^2\ge\dfrac{1}{4}\left(x+y\right)^2\)\(=\dfrac{1}{4}.1=\dfrac{1}{4}\)
Dấu "=" xảy ra khi :\(x=y=\dfrac{1}{2}\)
Vậy \(x^3+y^3\ge\dfrac{1}{4}\)
2.
a) Sửa đề: \(a^3+b^3\ge ab\left(a+b\right)\)
\(\Leftrightarrow\left(a^3-a^2b\right)+\left(b^3-ab^2\right)\ge0\)
\(\Leftrightarrow a^2\left(a-b\right)+b^2\left(b-a\right)\ge0\)
\(\Leftrightarrow\left(a-b\right)\left(a^2-b^2\right)\ge0\)
\(\Leftrightarrow\left(a-b\right)^2\left(a+b\right)\ge0\) (luôn đúng vì \(a,b\ge0\))
Đẳng thức xảy ra \(\Leftrightarrow a=b\)
b) Lần trước mk giải rồi nhá
3.
a) Áp dụng BĐT Cauchy-Schwarz dạng Engel\(P=\dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1}\ge\dfrac{\left(1+1+1\right)^2}{\left(x+y+z\right)+3}=\dfrac{9}{3+3}=\dfrac{3}{2}\)
Đẳng thức xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{x+1}=\dfrac{1}{y+1}=\dfrac{1}{z+1}\\x+y+z=3\end{matrix}\right.\Leftrightarrow x=y=z=1\)
b) \(Q=\dfrac{x}{x^2+1}+\dfrac{y}{y^2+1}+\dfrac{z}{z^2+1}\le\dfrac{x}{2\sqrt{x^2.1}}+\dfrac{y}{2\sqrt{y^2.1}}+\dfrac{z}{2\sqrt{z^2.1}}\)
\(=\dfrac{x}{2x}+\dfrac{y}{2y}+\dfrac{z}{2z}=\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}=\dfrac{3}{2}\)
Đẳng thức xảy ra \(\Leftrightarrow x^2=y^2=z^2=1\Leftrightarrow x=y=z=1\)
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\(\text{Xét hiệu:}\)
\(\frac{1}{x}+\frac{1}{y}-\frac{4}{x+y}=\frac{y.\left(x+y\right)}{xy.\left(x+y\right)}+\frac{x.\left(x+y\right)}{xy.\left(x+y\right)}-\frac{4xy}{xy.\left(x+y\right)}\)
\(=\frac{y^2+xy}{x^2y+xy^2}+\frac{x^2+xy}{x^2y+xy^2}-\frac{4xy}{x^2y+xy^2}\)
\(=\frac{x^2-2xy+y^2}{x^2y+xy^2}=\frac{\left(x-y\right)^2}{x^2y+xy^2}\)
\(\text{Vì }\left(x-y\right)^2\ge0\text{ với mọi x;y và }x>0;y>0\)
\(\text{nên: }\frac{\left(x-y\right)^2}{x^2y+xy^2}\ge0\text{ với mọi x;y hay }\frac{1}{x}+\frac{1}{y}-\frac{4}{x+y}\ge0\text{ với mọi x;y}\)
\(\Leftrightarrow\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\text{ với mọi x;y}\)
\(\text{Xét hiệu:}\)
\(\frac{1}{x}+\frac{1}{y}-\frac{4}{x+y}=\frac{y.\left(x+y\right)}{xy.\left(x+y\right)}+\frac{x.\left(x+y\right)}{xy.\left(x+y\right)}-\frac{4xy}{xy.\left(x+y\right)}\)
\(=\frac{xy+y^2}{xy.\left(x+y\right)}+\frac{x^2+xy}{xy.\left(x+y\right)}-\frac{4xy}{xy.\left(x+y\right)}=\frac{x^2-2xy+y^2}{xy.\left(x+y\right)}=\frac{\left(x-y\right)^2}{xy.\left(x+y\right)}\)
\(\text{Vì }\left(x-y\right)^2\ge0\text{ với mọi x;y };x>0;y>0\)
\(\text{Nên }\frac{\left(x-y\right)^2}{xy.\left(x+y\right)}\ge0\text{ với mọi x;y}\)
\(\text{hay }\frac{1}{x}+\frac{1}{y}-\frac{4}{x+y}\ge0\text{ với mọi x;y }\Leftrightarrow\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\text{ với mọi x;y}\)
\(\text{Dấu "=" xảy ra khi x=y}\)
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Bài 4 nha
Áp dụng BĐT cô si ta có
\(\frac{1}{x^2}+x+x\ge3\sqrt[3]{\frac{1}{x^2}.x.x}=3.\)
Tương tự với y . \(A\ge6\)dấu = xảy ra khi x=y=1
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1: \(x^2+x+1\)
\(=x^2+x+\dfrac{1}{4}+\dfrac{3}{4}\)
\(=\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}>0\forall x\)
2: \(2x^2+2x+1\)
\(=2\left(x^2+x+\dfrac{1}{2}\right)\)
\(=2\left(x^2+x+\dfrac{1}{4}+\dfrac{1}{4}\right)\)
\(=2\left(x+\dfrac{1}{2}\right)^2+\dfrac{1}{2}>0\forall x\)
3:
\(x^2+y^2=\left(x-y\right)^2+2xy=7^2+2\cdot60=169\)
\(x^4+y^4=\left(x^2+y^2\right)^2-2\cdot\left(xy\right)^2\)
\(=169^2-2\cdot60^2=21361\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\left(2a-3\right)\left(\frac{3}{4}a+1\right)=0\)
\(\Leftrightarrow\left[\begin{array}{nghiempt}2a-3=0\\\frac{3}{4}a+1=0\end{array}\right.\)
\(\Leftrightarrow\left[\begin{array}{nghiempt}2a=3\\\frac{3}{4}a=-1\end{array}\right.\)
\(\Leftrightarrow\left[\begin{array}{nghiempt}a=\frac{3}{2}\\a=-\frac{4}{3}\end{array}\right.\)
\(\left(2a-3\right)\left(\frac{3}{4}a+1\right)=0\)
<=> \(\left[\begin{array}{nghiempt}2a-3=0\\\frac{3}{4}a+1=0\end{array}\right.\)
<=> \(\left[\begin{array}{nghiempt}a=\frac{3}{2}\\a=-\frac{4}{3}\end{array}\right.\)
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\(4\left(x+y\right)\left(y+z\right)\left(1-y\right)\le\left(x+2y+z\right)^2\left(1-y\right)\)
\(\le\frac{1}{4}\left(x+2y+z\right)\left(x+2y+z+1-y\right)^2=x+2y+z\)
Dấu "=" xảy ra khi \(\hept{\begin{cases}x=z=\frac{1}{2}\\y=0\end{cases}}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Bài 3:
Áp dụng bất đẳng thức AM - GM có:
\(x+y+z+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\ge2\sqrt{x.\dfrac{1}{x}}+2\sqrt{y.\dfrac{1}{y}}+2\sqrt{z.\dfrac{1}{z}}\)
\(=2+2+2=6\)
Dấu " = " khi x = y = z = 1
Vậy...
3. Với x,y,z>0 áp dụng BĐT Cauchy ta có
\(x+y+z+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\)
\(=\left(x+\dfrac{1}{x}\right)+\left(y+\dfrac{1}{y}\right)+\left(z+\dfrac{1}{z}\right)\)
\(\ge2\sqrt{x.\dfrac{1}{x}}+2\sqrt{y.\dfrac{1}{y}}+2\sqrt{z.\dfrac{1}{z}}=2+2+2=6\)
Đẳng thức xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{1}{x}\\y=\dfrac{1}{y}\\z=\dfrac{1}{z}\end{matrix}\right.\Leftrightarrow x=y=z=1\)
1. Với a=b=c=0, ta thấy BĐT trên đúng
Với a,b,c>0 áp dụng BĐT Cauchy cho 3 số dương
\(a^3+a^3+b^3\ge3\sqrt[3]{a^3.a^3.b^3}=3\sqrt[3]{a^6b^3}=3a^2b\) (1)
\(b^3+b^3+c^3\ge3\sqrt[3]{b^3.b^3.c^3}=3\sqrt[3]{b^6c^3}=3b^2c\) (2)
\(c^3+c^3+a^3\ge3\sqrt[3]{c^3.c^3.a^3}=3\sqrt[3]{c^6a^3}=3c^2a\) (3)
Cộng (1), (2), (3) vế theo vế:
\(a^3+b^3+c^3\ge a^2b+b^2c+c^2a>\dfrac{a^2b+b^2c+c^2a}{3}\) (vì a,b,c>0)
Do đó BĐT trên đúng \(\forall a,b,c\ge0\)
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Bài 1:
a) \(x^2-2xy-25+y^2\) (Sửa đề)
\(=x^2-2xy+y^2-25\)
\(=\left(x-y\right)^2-5^2\)
\(=\left(x-y-5\right)\left(x-y+5\right)\)
Vậy ...
b) \(x\left(x-1\right)+y\left(1-x\right)\)
\(=x\left(x-1\right)-y\left(x-1\right)\)
\(=\left(x-1\right)\left(x-y\right)\)
Vậy ...
c) \(7x+7y-\left(x+y\right)\) (Sửa đề)
\(=7\left(x+y\right)-\left(x+y\right)\)
\(=\left(x+y\right)\left(7-1\right)\)
\(=6\left(x+y\right)\)
Vậy ...
d) \(x^4+y^4\)
\(=\left(x^2\right)^2+\left(y^2\right)^2\)
\(=\left(x^2+y^2\right)^2-2x^2y^2\)
\(=\left(x^2+y^2-\sqrt{2}xy\right)\left(x^2+y^2+\sqrt{2}xy\right)\)
Vậy ...
Bạn kia sai rồi
x > 0 ; y > 0 thì chưa chắc \(x\ge1;y\ge1\) được
Mình giải các bạn tham khảo nhé :
\(A=\left(x+1\right)\left(y+1\right)=x\left(y+1\right)+\left(y+1\right)=xy+x+y+1\)
\(=1+x+y+1=2+x+y\)
Ta lại có : \(x+y\ge2\sqrt{xy}=2.1=2\) ( bất đẳng thức cosi )
Dấu "=" xảy ra <=> \(x=y\)
\(\Rightarrow2+x+y\ge2+2=4\)
\(\Rightarrow A\ge4\) (Đpcm)
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