\(P=\frac{xy}{x+y+2}=\frac{\left(x+y\right)^2-\left(x^2+y^2\right)}{2\left(x+y+2\right)}=\frac{\left(x+y\right)^2-4}{2\left(x+y+2\right)}\)
\(=\frac{\left(x+y+2\right)\left(x+y-2\right)}{2\left(x+y+2\right)}=\frac{x+y-2}{2}\)
mặt khác ta có :
\(x+y\le\sqrt{2\left(x^2+y^2\right)}=\sqrt{2\cdot4}=2\sqrt{2}\)
\(P\le\frac{2\sqrt{2}-2}{2}=\sqrt{2}-1\)
dấu băng xảy ra khi \(x=y=\sqrt{2}\)
cho x,y,z la cac so thuc duong thoa man x+y+z=1 tim min A=x^3/(x^2+xy+y^2)+y^3/(y^2+yz+z^2)+z^3/(z^2+zx+x^2)
cho x,y la cac so duong thay doi va thoa man dieu kien x+y\(\le\)1. tim gia tri nho nhat cua bieu thuc M=\(\frac{1}{x^2+y^2}+\frac{1}{xy}+4xy\)
Cho 2 so thuc x , y thoa man x^2 + 4y^2 = 8
Tim max cua M=y ( 2 x - 3 y)
a)Tim cap (x,y) nguyen duong thoa man xy=3(y-x)
b)cho 2 so x,y >0 thoa man x+y = 1
Tim GTNN cua M=(x^2+1/y^2)(y^2+1/x^2)
a)Tim tat ca cac so nguyen duong x, y , z thoa man: \(\frac{x+y\sqrt{2013}}{y+z\sqrt{2013}}\)la so huu ti, dong thoi x2 + y2+ z2 la so nguyen to.
b) Tim so tu nhien x, y thoa man: x(1+x+x2) = y(y-1).
Cho x,y la cac so thuc duong. Tim gia tri nho nhat cua bieu thuc:
\(P=\frac{xy}{x^2+y^2}+\left(\frac{1}{x}+\frac{1}{y}\right)\sqrt{2\left(x^2+y^2\right)}\)
Cho hai so Thưc duong x, y thoa man x>=2y.Tim gia tri nho nhat cua bieu thuc P=(2x^2+y^2-2xy):xy
Cho x,y,z la cac so thuc duong thoa man xyz=2
Chung minh rang:\(\frac{x}{2x^2+y^2+5}+\frac{2y}{6y^2+z^2+6}+\frac{4z}{3z^2+4x^2+16}\le\frac{1}{2}\)
