\(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}< \frac{1}{100}+\frac{1}{100}+...+\frac{1}{100}=50.\frac{1}{100}=\frac{1}{2}< \frac{5}{6}\)
Vậy \(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}< \frac{5}{6}\)
Cho a, b, c, d dương. CM:
1) \(\frac{a^2}{b^5}+\frac{b^2}{c^5}+\frac{c^2}{d^5}+\frac{d^2}{a^5}\ge\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\)
2) \(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge\frac{a+b+c}{\sqrt[3]{abc}}\)
3) \(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{d^2}+\frac{d^2}{a^2}\ge\frac{a+b+c+d}{\sqrt[4]{abcd}}\)
4) \(\frac{1}{a^2+2bc}+\frac{1}{b^2+2ac}+\frac{1}{c^2+2ab}\ge9;a+b+c\le1\)
Cho a, b, c > 0 thỏa mãn abc = 1. Tìm GTLN
P = \(\frac{1}{\sqrt{a^5+b^2+ab+6}}+\frac{1}{\sqrt{b^5+c^2+bc+6}}\frac{1}{\sqrt{c^5+a^2+ac+6}}\)
Cm:
Nếu x,y,z >0 thỏa mãn
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=4\)
thì \(\frac{1}{2x+y+z}+\frac{1}{x+2y+z}+\frac{1}{x+y+2z}\le1\)
cho x,y,z > 0. Cmr: \(\frac{2x}{x^6+y^4}+\frac{2y}{y^6+z^4}+\frac{2z}{z^6+x^4}\le\frac{1}{x^4}+\frac{1}{y^4}+\frac{1}{z^4}\)
Cho a,b,c\(\ge\)0 và \(\frac{a}{1+a}+\frac{2b}{1+b}+\frac{3c}{1+c}\le1\)
Cmr : \(ab^2c^3\le\frac{1}{5^6}\)
cho \(z\ge y\ge x\ge0.CM:\)
\(y\left(\frac{1}{x}+\frac{1}{z}\right)+\frac{1}{y}\left(x+z\right)\le\left(x+z\right)\left(\frac{1}{x}+\frac{1}{z}\right)\)
C/m các BĐT sau :
\(1.a^3-3a+4\ge b^3-3b
\)
\(2,\frac{1}{\frac{1}{a+c}+\frac{1}{b+d}}\ge\frac{1}{\frac{1}{a}+\frac{1}{b}}+\frac{1}{\frac{1}{c}+\frac{1}{d}}\) với a, b, c, d>0
\(3,a^3+b^3\ge\frac{1}{4};a+b\ge1\)
4, \(a^3+b^3\le a^4+b^4;a+b\ge2\)
5, \(\left(a+b\right)\left(a^3+b^3\right)\left(a^5+b^5\right)\le4\left(a^9+b^9\right);a,b\ge0\)
6, \(\frac{c+a}{\sqrt{a^2+c^2}}\ge\frac{c+b}{\sqrt{c^2+b^2}};a>b>0,c>\sqrt{ab}\)
Các bn làm đc bài nào thì giúp mk với, cảm ơn ạ !
giải các bất đẳng thức sau:
1) \(\frac{3x-4}{x-2}\)>1
2) \(\frac{2x-5}{2-x}\)≤ -1
3) \(\frac{x^2+x-3}{x^2-4}\)≤ 1
4) \(\frac{2x^2-4x+3}{x^2-x-6}+\frac{1}{2}\)> 0
5) \(\frac{x^2+3x+2}{2x+3}\)≥\(\frac{2x-5}{4}\)
giúp mình với mng!!!
Bài 1: tính
a, 3\(\frac{1}{117}\). 4\(\frac{1}{119}\)-1\(\frac{116}{117}\).5\(\frac{118}{119}\)-\(\frac{5}{119}\)
b, 2\(\frac{1}{315}\).\(\frac{1}{651}\)-\(\frac{1}{105}\).3\(\frac{650}{651}\)-\(\frac{4}{315.651}\)+\(\frac{4}{105}\)
Làm giúp mình với mình cần luôn và ngay