Bình phương 2 vế BĐT đã cho:
\(\Leftrightarrow\left(x+y\right)^2\le\left(\left|x\right|+\left|y\right|\right)^2\)
\(\Leftrightarrow x^2+y^2+2xy\le x^2+y^2+2\left|xy\right|\)
\(\Leftrightarrow xy\le\left|xy\right|\) (luôn đúng)
Dấu "=" xảy ra khi \(xy\ge0\)
Cho \(x,y,z\in\left[2018,2019\right]\)
Tìm max của \(f\left(x,y,z\right)=\frac{\left|2018.2019-xy\right|}{\left(x+y\right)z}+\frac{\left|2018.2019-yz\right|}{\left(y+z\right)x}+\frac{\left|2018.2019-zx\right|}{\left(z+x\right)y}\)
Áp dụng BĐT Cô-si để tìm Max
a. \(y=\left(x+3\right)\left(5-x\right),\left(-3\le x\le5\right)\)
b. \(y=x\left(6-x\right)\left(0\le x\le6\right)\)
c. \(y=\left(x+3\right)\left(5-2x\right)\left(-3\le x\le\frac{5}{2}\right)\)
d. \(y=\left(2x+5\right)\left(5-2x\right)\left(-\frac{5}{2}\le x\le5\right)\)
e. \(y=\left(6x+3\right)\left(5-2x\right)\left(-\frac{1}{2}\le x\le\frac{5}{2}\right)\)
f. \(y=\frac{x}{x^2+2},x\ge0\)
g. \(y=\frac{x^2}{\left(x^2+2\right)^3}\)
Cho x,y \(\in R\) . Tìm Min của:
\(P=\sqrt{\left(x+1\right)^2+\left(y-1\right)^2}\sqrt{\left(x-1\right)^2+\left(y+1\right)^2}+\sqrt{\left(x+2\right)^2+\left(y+2\right)^2}\)
Cho x, y \(\in R\) . Tìm Min:
\(P=\sqrt{\left(x+1\right)^2+\left(y-1\right)^2}+\sqrt{\left(x-1\right)^2+\left(y+1^2\right)}+\sqrt{\left(x+2\right)^2+\left(y+2\right)^2}\)
cho x,y,z là số thực dương thỏa mãn xy+yz+xz=xyz
cmr \(\dfrac{xy}{z^3\left(1+x\right)\left(1+y\right)}+\dfrac{yz}{x^3\left(1+y\right)\left(1+z\right)}+\dfrac{xz}{y^3\left(1+x\right)\left(1+z\right)}\ge\dfrac{1}{16}\)
cho 3 số thực dương z;y;z thỏa mãn x+y+z<hoạc = 3/2
tìm GTNN của biểu thức :
\(P=\frac{z\left(xy+1\right)^2}{y^2\left(yz+1\right)}+\frac{x\left(yz+1\right)^2}{z^2\left(xz+1\right)}+\frac{y\left(xz+1\right)^2}{x^2\left(xy+1\right)}\)
cho \(z\ge y\ge x\ge0.CM:\)
\(y\left(\frac{1}{x}+\frac{1}{z}\right)+\frac{1}{y}\left(x+z\right)\le\left(x+z\right)\left(\frac{1}{x}+\frac{1}{z}\right)\)
\(\left\{{}\begin{matrix}\left(x-1\right)\sqrt{y}+\left(y-1\right)\sqrt{x}=2\sqrt{xy}\\x\sqrt{y-1}+y\sqrt{x}-1=xy\end{matrix}\right.\)
giải kiểu bất đẳng thức ạ . thanks
Cho \(x,y,z\) là các số thực dương. CMR: \(\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)\left(1+\frac{z}{x}\right)\ge2+\frac{2\left(x+y+z\right)}{3\sqrt{xyz}}\)
