Chứng minh bất đẳng thức
Cho x, y, z là các số dương (chứng minh hộ mình phần b) thôi)
a) CMR : \(3\left(x^2+y^2+z^2\right)\ge\left(x+y+z\right)^2\)
b) Cho x, y, z thỏa mãn : \(3+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=12\left(\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}\right)\)
CMR : \(\dfrac{1}{4x+y+z}+\dfrac{1}{x+4y+z}+\dfrac{1}{x+y+4z}\le\dfrac{1}{6}\)
Cho x+y+z=0; x,y,z\(\ne\)0. Chứng minh rằng:
\(\sqrt{\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}}\) = \(\left|\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right|\)
Cho các số thực x, y, z thỏa mãn \(7\left(\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}\right)=6\left(\dfrac{1}{xy}+\dfrac{1}{yz}+\dfrac{1}{zx}\right)=2016\).
Tìm max: \(P=\dfrac{1}{\sqrt{3\left(2x^2+y^2\right)}}+\dfrac{1}{\sqrt{3\left(2y^2+z^2\right)}}+\dfrac{1}{\sqrt{3\left(2z^2+x^2\right)}}\)
Cho x,y,z>0 thỏa mãn xyz=1. Tìm min \(P=\dfrac{x^2\left(y+z\right)}{y\sqrt{y}+2z\sqrt{z}}+\dfrac{y^2\left(z+x\right)}{z\sqrt{z}+2x\sqrt{x}}+\dfrac{z^2\left(x+y\right)}{x\sqrt{x}+2y\sqrt{y}}\)
1 Giải hệ pt \(\left\{{}\begin{matrix}\left(x-1\right)^3=1-\dfrac{27}{y^3}\\x^2+\dfrac{9}{y^2}=2x\end{matrix}\right.\)
2 CM \(n^4-10n^2+9\) chia hết 384 với mọi n lẻ
3 cho \(0\le x\le\dfrac{1}{2}\) tìm Max Q=\(x^2\left(1-2x\right)\)
4 cho x,y,z dương thỏa \(x^2+y^2+z^2=3xyz\).CM \(\dfrac{x^2}{x^4+yz}+\dfrac{y^2}{y^4+xz}+\dfrac{z^2}{z^4+xy}\le\dfrac{3}{2}\)
Giải hệ phương trình :
a) \(\left\{{}\begin{matrix}x^2+y^2=1\\x^2+y^2=1\end{matrix}\right.\)
b)\(\left\{{}\begin{matrix}\sqrt{x}+\sqrt{y}+\sqrt{z}=2014\\\dfrac{1}{3x+2y}+\dfrac{1}{3y+2z}+\dfrac{1}{3z+2x}=\dfrac{1}{x+2y+3z}+\dfrac{1}{y+2x+3x}+\dfrac{1}{z+2x+3y}\end{matrix}\right.\)
1.Cho x, y \(\ge\)0 và x+ y=1
Chứng minh rằng : \(x^3+y^3\ge\dfrac{1}{4}\)
2. Cho \(a,b,c\ge0\).Chứng minh rằng:
a, \(a^3+b^3>ab\left(a+b\right)\)
b, \(a^3+b^3+c^3\ge a^2b+ b^2c+c^2a\)
3. Cho x+ y+ z=3 và x, y, z>0. Chứng minh rằng:
a, \(P=\dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1}\ge\dfrac{3}{2}\)
b, \(Q=\dfrac{x}{x^2+1}+\dfrac{y}{y^2+1}+\dfrac{z}{z^2+1}\le\dfrac{3}{2}\)
Cho x;y>0
CMR: \(\dfrac{\left(x^3+8\right)\left(y^2-y+1\right)}{\left(x^2+x\right)\left(xy^2+2\right)}\ge\dfrac{1}{2}\)
Tìm \(x;y\in N\)tmãn : \(\sqrt{x}+\sqrt{y}=\sqrt{2012}\)
2, Rút gọn bt
\(P=\dfrac{x}{x-\sqrt{x}}+\dfrac{2}{x+2\sqrt{x}}+\dfrac{x+2}{\left(\sqrt{x}-1\right)\left(x+2\sqrt{x}\right)}\)
b, gpt : \(x^2-2x-x\sqrt{x}-2\sqrt{x}+4=0\)
3, cho x>1 ; y>0 , cm
\(\dfrac{1}{\left(x+1\right)^3}+\left(\dfrac{x-1}{y}\right)^3+\dfrac{1}{y^3}\ge3\left(\dfrac{3-2x}{x-1}+\dfrac{x}{y}\right)\)
Unruly Kid
