1.Cho x,y > 0 và x^2 + y^2 = 1
Tìm GTNN của \(A=\frac{-2xy}{1+xy}\)
2.cho các số dương x, y,z thỏa man x+y+z=4. Chứng minh \(\frac{1}{xy}+\frac{1}{xz}>=1\)
3.3)cho các số x, y không âm thỏa mãn x+y=1 . tìm gtnn ,gtln của A =x^2+y^2
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\(\frac{x^2}{y+1}+\frac{y+1}{4}\ge x;\frac{y^2}{z+1}+\frac{z+1}{4}\ge y;\frac{z^2}{x+1}+\frac{x+1}{4}\ge z\)
\(\Rightarrow VT\ge\frac{3}{4}\left(x+y+z\right)-\frac{3}{4}\ge\frac{3}{4}.2=\frac{3}{2}\)
2. Có : 1/x + 1/y + 1/z = 0
=> 1 + x/y + x/z = 0 => x/y + x/z = -1
Tương tự : y/x + y/z = -1 ; z/x + z/y = -1
=> x/y + x/z + y/x + y/z + z/x + z/y = -3
Lại có : 1/x+1/y+1/z = 0
<=> xy+yz+zx/xyz = 0
<=> xy+yz+zx = 0
Xét : 0 = (xy+yz+zx).(1/x^2+1/y^2+1/z^2)
= xy/z^2+xz/y^2+xy/z^2+x/y+y/x+y/z+z/y+z/x+x/z
= xy/z^2+xz/y^2+xy/z^2-3
=> xy/z^2+xz/y^2+xy/z^2 = 3
=> ĐPCM
Tk mk nha
Áp dụng BĐT Cô si ta có:
\(1=\left(a+b+c\right)^2\ge4a\left(b+c\right)\)
\(\Leftrightarrow b+c\ge4a\left(b+c\right)^2\)
Mà \(\left(b+c\right)^2\ge4bc\)
\(\Rightarrow b+c\ge4a.4bc=16abc\)
Bạn tham khảo tại đây:
Câu hỏi của trieu dang - Toán lớp 8 - Học toán với OnlineMath
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)
\(\Rightarrow\frac{\left(yz+xz+xy\right)}{xyz}=0\)
\(\Rightarrow yz+zx+xy=0\)
Ta có : \(x^2+2yz=x^2+yz+yz\)
\(=x^2+yz-zx-xy\)
\(=x\left(x-z\right)-y\left(x-z\right)\)
\(=\left(x-y\right)\left(x-z\right)\)
Tương tự : \(y^2+2xz=y^2+xz+xz\)
\(=y^2+xz-xy-yz\)
\(=y\left(y-x\right)+z\left(x-y\right)\)
\(=\left(x-y\right)\left(z-y\right)\)
\(z^2+2xy=\left(x-z\right)\left(y-z\right)\)
\(\Rightarrow M=\frac{yz}{\left(x-y\right)\left(x-z\right)}+\frac{xz}{\left(x-y\right)\left(z-y\right)}+\frac{xy}{\left(x-z\right)\left(y-z\right)}\) \(M=\frac{yz\left(y-z\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}-\frac{xz\left(x-z\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}+\frac{xy\left(x-y\right)}{\left(x-z\right)\left(y-z\right)\left(x-y\right)}\)
\(M=\frac{yz\left(y-z\right)-xz\left(x-z\right)+xy\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}=\frac{yz\left(y-z\right)-xz\left(x-y+y-z\right)+xy\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(A=\frac{\left(yz-xz\right)\left(y-z\right)+\left(xy-xz\right)\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}=\frac{\left(x-y\right)\left(x-z\right)\left(y-z\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}=1\)
Áp dụng BĐT \(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\)với a,b>0
Ta có: \(\frac{4xy}{z+1}=\frac{4xy}{2z+x+y}\le\frac{xy}{x+z}+\frac{xy}{y+z}\)
Tương tự: \(\frac{4yz}{x+1}\le\frac{yz}{x+y}+\frac{yz}{x+z}\)
\(\frac{4zx}{y+1}\le\frac{zx}{y+x}+\frac{zx}{y+z}\)
\(\Rightarrow4\left(\frac{xy}{z+1}+\frac{yz}{x+1}+\frac{zx}{y+1}\right)\le\frac{xy}{x+z}+\frac{xy}{y+z}+\frac{yz}{x+y}+\frac{yz}{x+z}+\frac{zx}{y+x}+\frac{zx}{y+z}=x+y+z=1\)
\(\Rightarrow\frac{xy}{z+1}+\frac{yz}{x+1}+\frac{zx}{y+1}\le\frac{1}{4}\)
Dấu "=" xảy ra khi: x=y=z>0
Bài 2:
+) Với y=0 <=> x=0
Ta có: 1-xy= 12 (đúng)
+) Với \(y\ne0\)
Ta có: \(x^6+xy^5=2x^3y^2\)
\(\Leftrightarrow x^6-2x^3y^2+y^4=y^4-xy^5\)
\(\Leftrightarrow\left(x^3-y^2\right)^2=y^4\left(1-xy\right)\)
\(\Rightarrow1-xy=\left(\frac{x^3-y^2}{y^2}\right)^2\)
Ta có \(\frac{x+2xy+1}{x+xy+xz+1}=\frac{x+2xy+xyz}{x+xy+xz+xyz}=\frac{1+2y+yz}{\left(y+1\right)\left(z+1\right)}\)
Tương tự => \(M=\frac{1+2y+yz}{\left(y+1\right)\left(z+1\right)}+\frac{1+2z+zx}{\left(1+x\right)\left(z+1\right)}+\frac{1+2x+xy}{\left(1+x\right)\left(y+1\right)}\)
=> \(M=\frac{\left(1+2y+yz\right)\left(1+x\right)+\left(1+2z+zx\right)\left(1+y\right)+\left(1+2x+xy\right)\left(1+z\right)}{\left(1+x\right)\left(1+y\right)\left(1+z\right)}\)
=>\(M=\frac{6+3\left(x+y+z\right)+3\left(xy+yz+xz\right)}{2+\left(x+y+z\right)+\left(xy+yz+xz\right)}=3\)
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