a)\(S=\dfrac{a+b}{c}+\dfrac{b+c}{a}+\dfrac{c+a}{b}=\left(\dfrac{a}{c}+\dfrac{c}{a}\right)+\left(\dfrac{b}{a}+\dfrac{a}{b}\right)+\left(\dfrac{c}{b}+\dfrac{b}{c}\right)\)

Áp dụng BĐT cosi:

\(\dfrac{a}{c}+\dfrac{c}{a}\ge2\sqrt{\dfrac{ac}{ca}}=2\)

\(\dfrac{a}{b}+\dfrac{b}{c}\ge2\)

\(\dfrac{b}{c}+\dfrac{c}{b}\ge2\)

=>S\(\ge\)6

Dấu = xảy ra khi và chỉ khi \(\left\{{}\begin{matrix}\dfrac{a}{c}=\dfrac{c}{a}\\\dfrac{a}{b}=\dfrac{b}{a}\\\dfrac{c}{b}=\dfrac{b}{c}\end{matrix}\right.\)<=>a=b=c

b)S\(\ge\)6

=>GTNN của S=6 xảy ra khi a=b=c

Đề có bị sao không vậy? \(S\) không thể bằng \(2\) Sửa đề:

Chứng minh rằng \(S\ge6\)

Giải:

Ta có:

\(S=\dfrac{a+b}{c}+\dfrac{b+c}{a}+\dfrac{a+c}{b}\)

\(=\left(\dfrac{a}{c}+\dfrac{b}{c}\right)+\left(\dfrac{b}{a}+\dfrac{c}{a}\right)+\left(\dfrac{a}{b}+\dfrac{c}{b}\right)\)

\(=\left(\dfrac{a}{c}+\dfrac{c}{a}\right)+\left(\dfrac{b}{c}+\dfrac{c}{b}\right)+\left(\dfrac{b}{a}+\dfrac{a}{b}\right)\)

\(\Rightarrow S\ge2+2+2=6\)

Vậy \(S\ge6\) (Đpcm)

đề k bị sao bn ơi

a) \(\frac{6}{7}\) và \(\frac{11}{10}\)

\(\frac{6}{7}< 1\)

\(\frac{11}{10}>1\)

\(\Rightarrow\frac{6}{7}< 1< \frac{11}{10}\Rightarrow\frac{6}{7}< \frac{11}{10}\)

b) \(\frac{-5}{17}\) và \(\frac{2}{7}\)

\(\frac{-5}{17}< 0\)

\(\frac{2}{7}>0\)

\(\Rightarrow\frac{-5}{17}< 0< \frac{2}{7}\)\(\Rightarrow\frac{-5}{17}< \frac{2}{7}\)

c) \(\frac{419}{-723}\) và \(\frac{-697}{-313}\)

\(\frac{419}{-724}< 0\)

\(\frac{-697}{-313}>0\)

\(\Rightarrow\frac{419}{-724}< 0< \frac{-697}{-313}\Rightarrow\frac{419}{-723}< \frac{-697}{-313}\)

a) \(\left\{{}\begin{matrix}\dfrac{11}{10}>\dfrac{10}{10}\\\dfrac{10}{10}=\dfrac{7}{7}>\dfrac{6}{7}\end{matrix}\right.\) \(\Rightarrow\dfrac{11}{10}>\dfrac{6}{7}\)

b) \(\left\{{}\begin{matrix}\dfrac{2}{7}>\dfrac{0}{7}\\\dfrac{0}{7}=\dfrac{0}{17}>-\dfrac{5}{17}\end{matrix}\right.\) \(\Rightarrow\dfrac{2}{7}>\dfrac{-5}{17}\)

c)

\(\left\{{}\begin{matrix}\dfrac{-697}{-313}=\dfrac{697}{313}>\dfrac{0}{313}\\\dfrac{0}{313}=\dfrac{0}{723}>\dfrac{-419}{723}=\dfrac{419}{-723}\end{matrix}\right.\) \(\Rightarrow\dfrac{-697}{-313}>\dfrac{419}{-723}\)

Mk chỉ cho gợi ý thôi nha.

Cũng không khó lắm nên bạn có thể tự làm mà.

a) So sánh với 1

b) So sánh với 0

c) So sánh với 0

Chúc bạn học tốt!

Vì 18/91 < 18/90 =1/5

23/114>23115=1/5

vậy 18/91<1/5<23/114

suy ra 18/91<23/114

vì 21/52=210/520

Mà 210/520=1-310/520

213/523=1-310/523

310/520>310/523

vậy 210/520<213/523

suy ra 21/52<213/523

choáng

dài quá mik ko làm âu

\(\dfrac{b}{c}=\dfrac{3}{4}\)

Mấy bài dễ u tự giải quyết nha

3) \(\dfrac{2013}{2014}+\dfrac{2014}{2015}+\dfrac{2015}{2013}\)

\(=\left(1-\dfrac{1}{2014}\right)+\left(1-\dfrac{1}{2015}\right)+\left(1+\dfrac{2}{2013}\right)\)

\(=3+\dfrac{2}{2013}-\dfrac{1}{2014}-\dfrac{1}{2015}\)

\(=3+\left(\dfrac{1}{2013}-\dfrac{1}{2014}\right)+\left(\dfrac{1}{2013}-\dfrac{1}{2015}\right)>3\)