\(x^4+y^4\ge\dfrac{\left(x^2+y^2\right)^2}{2}\ge\dfrac{\left(\dfrac{\left(x+y\right)^2}{2}\right)^2}{2}=\dfrac{1}{8}\)
Dấu "=" xảy ra khi \(x=y=\dfrac{1}{2}\)
cho x>0,y>0va x+y=1 cm \(8\left(x^4+y^4\right)+\dfrac{1}{xy}\ge5\)
1) rút gọn biểu thức sau :
a) \(\dfrac{x+2\sqrt{x}-3}{\sqrt{x}-1}\) b) \(\dfrac{4y+3\sqrt{y}-7}{4\sqrt{y}+7}\) c ) \(\dfrac{x\sqrt{y}-y\sqrt{x}}{\sqrt{x}-\sqrt{y}}\)
d) \(\dfrac{x-3\sqrt{x}-4}{x-\sqrt{x}-12}\) e) \(\dfrac{1+\sqrt{x}+\sqrt{y}+\sqrt{xy}}{1+\sqrt{y}}\) ( với x>0 , y>0 )
f) \(\sqrt{8-2\sqrt{15}}+\sqrt{5}+\sqrt{3}\) g) \(\sqrt{9-2\sqrt{4}}-\sqrt{9+2\sqrt{14}}\)
Cho x,y,z >0 tm x+y+z=3
C/m :\(\dfrac{x^3}{y^3+8}+\dfrac{y^3}{z^3+8}+\dfrac{z^3}{x^3+8}\ge\dfrac{1}{9}+\dfrac{2}{27}\left(xy+yz+zx\right)\)
Cho biểu thức A= \(\dfrac{\left(x^2+y\right)\left(\dfrac{1}{4}+y\right)+x^2y^2+\dfrac{3}{4}\left(\dfrac{1}{3}+y\right)}{x^2y^2+1+\left(x^2-y\right)\left(1-y\right)}\)
a) Tìm đkxđ A
b) Chứng minh A không phụ thuộc vài x
c) Tìm GTNN của A
Câu 1 . Cho \(a,b\ge3.\) Chứng minh rằng
\(A=21\left(a+\dfrac{1}{b}\right)+3\left(b+\dfrac{1}{a}\right)\ge80\)
Câu 2. Giải phương trình :
\(x^2+6x-1=2\sqrt{5x^3-3x^2+3x-2}\)
Câu 3. Tìm GTNN của
\(Q=\dfrac{1}{2}\left(\dfrac{x^{10}}{y^2}+\dfrac{y^{10}}{x^2}\right)+\dfrac{1}{4}\left(x^{16}+y^{16}\right)-\left(1+x^2y^2\right)^2\)
Câu 4 . Giải phương trình
\(\dfrac{\sqrt{x-2009}-1}{x-2009}+\dfrac{\sqrt{y-2010}-1}{y-2010}+\dfrac{\sqrt{z-2011}-1}{z-2011}=\dfrac{3}{4}\)
giai he phuong trinh \(\left\{{}\begin{matrix}\dfrac{4}{x+y-1}-\dfrac{5}{2x-y+1}=\dfrac{5}{2}\\\dfrac{3}{x+y-1}-\dfrac{1}{2x-y+3}=\dfrac{7}{5}\end{matrix}\right.\)
Cho x,y,z>0. CM: \(\dfrac{xy}{z^2\left(x+y\right)}+\dfrac{yz}{x^2\left(y+z\right)}+\dfrac{zx}{y^2\left(z+x\right)}\ge\dfrac{1}{2}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)
Cho x2-x+y2-y=xy. CMR : (y-1)2≤\(\dfrac{4}{3}\)
4) Cho x,y > 0 ; x + y = 1 . Tìm min M = \(\dfrac{1}{x^2+y^2}+\dfrac{1}{xy}\)
