Ta có: \(P=1+\left(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\right)+\left(\frac{1}{a^3b^3}+\frac{1}{b^3c^3}+\frac{1}{a^3c^3}+\frac{1}{a^3b^3c^3}\right)\)
\(P\ge a+\frac{3}{abc}+\frac{3}{a^2b^2c^2}+\frac{1}{a^3b^3c^3}=\left(1+\frac{1}{abc}\right)^3\) (BĐT Cosi cho 3 số dương)
Theo BĐT Cosi \(abc\le\left(\frac{a+b+c}{3}\right)^3=8̸\)\(\Rightarrow abc\le8\Rightarrow\frac{1}{abc}\ge\frac{1}{8}\)
Vậy \(P\ge\left(1+\frac{1}{8}\right)^3=\frac{729}{512}\)
Dấu "=" xảy ra khi a=b=c=2
Cho a,b,c >0 t/m a+b+c=3
Tìm GTNN của \(P=\left(1+\frac{3}{a}\right)\left(1+\frac{3}{b}\right)\left(1+\frac{3}{c}\right)\)
a) Cho a,b>0, a+b=<1.Tìm Min của A = \(^{\left(a+\frac{1}{a}\right)^2}\)+ \(^{\left(b+\frac{1}{b}\right)^2}\)
b) Cho a,b,c >0, a+b+c =<1,5. Tìm Min của B= \(\left(a+\frac{1}{a}\right)^2+\left(b+\frac{1}{b}\right)^2+\left(c+\frac{1}{c}\right)^2\)
Cho a,b,c > 0, abc=1
CMR:
\(\frac{a^3}{\left(1+b\right)\left(1+c\right)}=\frac{b^3}{\left(1+c\right)\left(1+a\right)}=\frac{c^3}{\left(1+b\right)\left(1+a\right)}\)>=\(\frac{3}{4}\)
1.tìm các nghiem nguyen cua phuong trinh: 54x^3+1=y^3
2.cho x+y=1 và xy khac 0.chung mih \(\frac{x}{y^3-1}+\frac{y}{x^3-1}+\frac{2\left(x-y\right)}{x^2y^2+3}=0\)
3.cho a,b,c la cac so thuc duong.chung minh :\(\left(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\right)^2+\frac{14abc}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\ge4\)
Cho a,b,c>0 thỏa mãn \(a+b+c\le\frac{3}{2}\)Tìm GTNN: \(P=\left(3+\frac{1}{a}+\frac{1}{b}\right)\left(3+\frac{1}{b}+\frac{1}{c}\right)\left(3+\frac{1}{c}+\frac{1}{a}\right)\)
1)\(\frac{1}{\left(a-b\right)\left(a-c\right)}+\frac{1}{\left(b-c\right)\left(b-a\right)}+\frac{1}{\left(c-a\right)\left(c-b\right)}\)
2)\(\frac{a^2}{\left(a-b\right)\left(a-c\right)}+\frac{b^2}{\left(b-c\right)\left(b-a\right)}\frac{c^2}{\left(c-a\right)\left(c-b\right)}\)
3)\(\frac{1}{x^2+3x+2}+\frac{2x}{x^3+4x^2+4x}+\frac{1}{x^2+5x+6}\)
Cho ba số khác 0 thoả mãn \(\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=1\) . Tính giâ trị của biểu thức \(M=2020\left(a^3+b^3\right)\left(b^5+c^5\right)\left(c^7+a^7\right)+2019\)
Cho 3 số thực dương a;b;c thỏa mãn \(7\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)=6\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)+2015\)
Tìm GTLN của \(P=\frac{1}{\sqrt{3\left(2a^2+b^2\right)}}+\frac{1}{\sqrt{3\left(2b^2+c^2\right)}}+\frac{1}{\sqrt{3\left(2c^2+a^2\right)}}\)
Cho a,b,c>0 và abc=1. CMR:
\(\frac{1}{a^3\left(b+c\right)}+\frac{1}{b^3\left(c+a\right)}+\frac{1}{c^3\left(a+b\right)}\ge\frac{3}{2}\)
