Đặt \(\left\{{}\begin{matrix}2^a=x>0\\2^b=y>0\\2^c=z>0\end{matrix}\right.\) \(\Rightarrow xyz=2^a.2^b.2^c=2^{a+b+c}=1\)
BĐT cần c/m trở thành: \(x^3+y^3+z^3\ge x+y+z\) với \(xyz=1\)
Ta có:
\(x^3+1+1\ge3x\) ; \(y^3+1+1\ge3y\) ; \(z^3+1+1\ge3z\)
\(\Rightarrow x^3+y^3+z^3\ge\left(x+y+z\right)+2\left(x+y+z\right)-6\ge x+y+z+6-6=x+y+z\)
Bài 1 : Cmr :
a, \(a+\frac{1}{a-1}\ge3\) với mọi a>1
b, \(\frac{a^2+2}{\sqrt{a^2+1}}\ge2\) với mọi a \(\in R\)
Bài 2 : Cho a>0. Cmr \(\frac{a^2+5}{\sqrt{a^2+4}}\ge2\)
Bài 3 : Cho a,b,c>0. Cmr \(1< \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{a+c}< 2\)
Cho a,b,c > 0, abc = 1. CMR :
(a+b)(b+c)(c+a)\(\ge2\left(1+a+b+c\right)\)
(4)Bài 1:Với \(\forall\) a>b>0. CMR: a+ \(\frac{1}{b\left(a-b\right)}\ge3\)
(7) Bài 2: Cho a,b,c \(\ne\) 0 .CMR: \(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\)
(8) Bài 3: Cho a,b,c>0 thõa mãn abc=1
CMR: \(\frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}}\ge\sqrt{a}+\sqrt{b}+\sqrt{c}+3\)
Cho a>0 , b>0,c>0 . CMR : \(\dfrac{a^8+b^8+c^8}{a^3b^3c^3}\) \(\ge\) \(\dfrac{1}{a}\)+\(\dfrac{1}{b}\)+ \(\dfrac{1}{c}\)
Cho a,b,c>0 và abc=1 CMR \(\frac{a^4\left(b^2+c^2\right)}{b^3+2c^3}+\frac{b^4\left(a^2+c^2\right)}{c^3+2a^3}+\frac{c^4\left(a^2+b^2\right)}{a^3+2b^3}\ge2\)
Cho a,b,c không âm thỏa mãn \(ab+bc+ca+abc=4\).CMR
\(\sqrt{a^2+8}+\sqrt{b^2+8}+\sqrt{c^2+8}\le a+b+c+6\)
1/CMR
a/\(x^4-2x^3+2x^2-2x+1\ge0\forall x\in R\)
b/cho \(a\ge0;b\ge2;a+b+c=3\)
CMR: \(a^2+b^2+c^2\le5\)
c/ a,b,c>0 CMR: \(\frac{b+c}{a}+\frac{a+c}{b}+\frac{a+b}{c}\ge4\left(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\right)\)
Câu 1 ; a, Rút gọn A=\(\frac{\sqrt{5+\sqrt{5}-2\sqrt{2}\sqrt{3+\sqrt{5}}}}{\sqrt{3-\sqrt{5}}+\sqrt{2}}\)
b, cho \(\frac{a}{b+c}\frac{b}{a+c}\frac{c}{a+b}=1\) tính P=\(a^2+b^2+c^2+\frac{a^3}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\)
Câu 2 ; a, cho các số nguyên dương a,b ,c thỏa mãn \(\left(a-b\right)\left(a-c\right)\left(b-c\right)=a+b+c\) CM a+b+c chia hết cho 54
b, giải pt x2+7x +14-2\(\sqrt{x-4}\)=0
Câu 3 ; cho a,b,c >0 thỏa mãn \(\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}\ge2\) CMR abc\(\ge\) 8
ab + bc + ca = 3 ; a,b,c >0 CMR:
\(\frac{a}{a^2+7}+\frac{b}{b^2+7}+\frac{c}{c^2+7}\le\frac{3}{8} \)
