Áp dụng BĐT Cauchy-Schwarz:
\(P=\frac{1^2}{x}+\frac{2^2}{y}+\frac{3^2}{z}\ge\frac{\left(1+2+3\right)^2}{x+y+z}=\frac{36}{1}=36\)
Dấu "=" xảy ra \(\Leftrightarrow x=\frac{1}{6};y=\frac{1}{3};z=\frac{1}{2}\)
Cho x,y,z>0 và \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{16}{x+y+z}\).Tìm min \(P=\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\)
Cho x, y, z > 0 thoả mãn: x + y + z = 1. Tìm Min A = \(\left(x+\frac{1}{x}\right)^2+\left(y+\frac{1}{y}\right)^2+\left(z+\frac{1}{z}\right)^2\)
Tìm GTNN của các biểu thức sau:
1) Cho x,y >0
Tìm Min P= \(\frac{x+y}{\sqrt{xy}}+\frac{\sqrt{xy}}{x+y}\)
2) Cho x, y, z >0 và x+y+z ≤ \(\frac{3}{4}\)
Tìm Min P= \(\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{y}+\sqrt{z}\right)\left(\sqrt{z}+\sqrt{x}\right)\)+ \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)
3) Cho a,b >0 và a+b≥3
Tìm Min P=\(a+b+\frac{1}{2a}+\frac{2}{b}\)
1. gpt : \(\frac{2x+1}{\sqrt{x^2+2}}+\left(x+1\right)\sqrt{1+\frac{2x+1}{x^2+2}}+x=0\)
2. \(\left\{{}\begin{matrix}x,y,z>0\\x+y+z\le\frac{3}{2}\end{matrix}\right.\) Tìm min \(Q=\frac{x}{y^2z}+\frac{y}{z^2x}+\frac{z}{x^2y}+\frac{x^5}{y}+\frac{y^5}{z}+\frac{z^5}{x}\)
Cho x,y,z là các số thực dương thỏa mãn x+y+z=1
\(\text{Tìm Min }\text{của}\text{ }P=\frac{x+yz}{y+z}+\frac{y+zx}{z+x}+\frac{z+xy}{x+y}\)
2. a) \(\left\{{}\begin{matrix}x,y,z>1\\x+y+z=xyz\end{matrix}\right.\) Tìm min \(P=\frac{x-1}{y^2}+\frac{y-1}{z^2}+\frac{z-1}{x^2}\)
b) \(a,b,c>0.Cmr:\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^2\ge\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
c) \(\left\{{}\begin{matrix}x,y,z\ge0\\x^2+y^2+z^2=2\end{matrix}\right.\) Tìm max \(P=\frac{x^2}{x^2+yz+x+1}+\frac{y+z}{x+y+z+1}-\frac{1+yz}{9}\)
d) \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c=3\end{matrix}\right.\). Cmr: \(\frac{a}{ab+3c}+\frac{b}{bc+3a}+\frac{c}{ca+3b}\ge\frac{3}{4}\)
Cho x, y, z > 0 thỏa mãn \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=3\). CMR:
\(\frac{x}{x^4+1+2xy}+\frac{y}{y^4+1+2yz}+\frac{z}{z^4+1+2zx}\le\frac{3}{4}\)
Cho các số x,y,z không âm , không đồng thời bằng 0 và thỏa mãn \(\frac{1}{x+1}+\frac{1}{y+2}+\frac{1}{z+3}\le1\)
Tìm Pmin=\(x+y+z+\frac{1}{x+y+z}\)
1. a) Tìm \(n\in N\)*, \(n>2008\) sao cho \(2^{2008}+2^{2012}+2^{2013}+2^{2014}+2^{2016}+2^n\) là số chính phương
b) tìm x,y > 0 thỏa mãn \(x^2+y^2=2\left(x+y\right)\left(\sqrt{x}+\sqrt{y}-2\right)\)
2. a) \(\left\{{}\begin{matrix}a\ge0\\a+b\ge1\end{matrix}\right.\). Min \(A=\frac{8a^2+b}{4a}+b^2\)
b) \(\left\{{}\begin{matrix}a,b\ge0\\\left(a-b\right)^2=a+b+2\end{matrix}\right.\). Cmr: \(\left(1+\frac{a^3}{\left(b+1\right)^3}\right)\left(1+\frac{b^3}{\left(b+1\right)^3}\right)\le9\)
c) \(x,y>0;\left(x+\sqrt{1+x^2}\right)\left(y+\sqrt{1+y^2}\right)=2020\). Min P = x + y
d) \(x,y,z>0;\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}=6\). Min \(P=\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}\)
e) \(\left\{{}\begin{matrix}x,y,z>0\\x+y+z+4xyz=4\end{matrix}\right.\) Cmr: \(\left(1+xy+\frac{y}{z}\right)\left(1+yz+\frac{z}{x}\right)\left(1+zx+\frac{x}{y}\right)\ge27\)
f) \(\left\{{}\begin{matrix}x,y,z\ge1\\3x^2+4y^2+5z^2=52\end{matrix}\right.\). Min P = x + y + z
g) \(x,y>0\). Min \(P=\frac{2}{\sqrt{\left(2x+y\right)^3+1}-1}+\frac{2}{\sqrt{\left(x+2y\right)^3+1}-1}+\frac{\left(2x+y\right)\left(x+2y\right)}{4}-\frac{8}{3\left(x+y\right)}\)
