Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
e)
\(\dfrac{a^2+b^2+c^2}{3}\ge\left(\dfrac{a+b+c}{3}\right)^2\)
\(\Leftrightarrow3\left(a^2+b^2+c^2\right)\ge a^2+b^2+c^2+2\left(ab+bc+ca\right)\)
\(\Leftrightarrow2\left(a^2+b^2+c^2\right)\ge2\left(ab+bc+ac\right)\)
\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2ac-2bc\ge0\)
\(\Leftrightarrow\left(a^2-2ab+b^2\right)+\left(a^2-2ac+c^2\right)+\left(b^2-2bc+c^2\right)\ge0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2\ge0\) ( luôn đúng)
=> ĐPCM
Mạn phép ko chép lại đề , mk làm luôn
a) \(D=\left[\dfrac{\left(\sqrt{a}+\sqrt{b}\right)\left(1+\sqrt{ab}\right)+\left(\sqrt{a}+\sqrt{b}\right)\left(1-\sqrt{ab}\right)}{1-ab}\right]:\dfrac{a+b+2ab+1-ab}{1-ab}\)\(D=\dfrac{\left(\sqrt{a}+\sqrt{b}\right)\left(1+\sqrt{ab}+1-\sqrt{ab}\right)}{1-ab}.\dfrac{1-ab}{a+b+ab+1}\)
\(D=\dfrac{2\left(\sqrt{a}+\sqrt{b}\right)}{\left(b+1\right)\left(a+1\right)}\)
D=A/B
a)
B=1+(a+b+2ab)/(1-ab)=(a+b+ab)/(1-ab)
dk: a,b≥0; a.b≠1
1/B=(1-ab)/(a+b+ab)
A=√a+√b)[(1+√ab)+(1-√ab)]/(1-ab)=2(√a+√b)/(1-ab)
D=2(√a+√b)/[(a+1)(b+1)]
b)
a=2/(√3+2)=2(2-√3)/[(2+√3)(2-√3)]=2(2-√3)=(√3-1)^2
a) \(B=\)\(\dfrac{\dfrac{1}{\sqrt{x}}+\dfrac{\sqrt{x}}{\sqrt{x}+1}}{\dfrac{\sqrt{x}}{x+\sqrt{x}}}\) ĐKXĐ: x>0
=\(\dfrac{\dfrac{\sqrt{x}+1+\sqrt{x}.\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+1\right)}}{\dfrac{\sqrt{x}}{x+\sqrt{x}}}\)
\(=\dfrac{x+\sqrt{x}+1}{x+\sqrt{x}}:\dfrac{\sqrt{x}}{x+\sqrt{x}}\)
=\(\dfrac{x+\sqrt{x}+1}{x+\sqrt{x}}\times\dfrac{x+\sqrt{x}}{\sqrt{x}}\)
\(=\dfrac{x+\sqrt{x}+1}{\sqrt{x}}\)
b)
Theo câu a ) ta có :
B=\(\dfrac{x+\sqrt{x}+1}{\sqrt{x}}\)
Xét : \(x+\sqrt{x}+1=x+2.\sqrt{x}.\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{3}{4}\)
=\(\left(\sqrt{x}+\dfrac{1}{2}\right)^2+\dfrac{3}{4}>0\) (với mọi x>0) (1)
Xét:
\(\sqrt{x}>0\) (2)
Từ (1) và (2) =>\(\dfrac{x+\sqrt{x}+1}{\sqrt{x}}>0\) (ĐPCM)
c) B=\(\dfrac{x+\sqrt{x}+1}{\sqrt{x}}\) ( theo câu a)
=\(\dfrac{x}{\sqrt{x}}+\dfrac{1}{\sqrt{x}}+1\)
=\(\sqrt{x}+\dfrac{1}{\sqrt{x}}+1\)
Áp dụng BĐT cô si cho \(\sqrt{x}\)và \(\dfrac{1}{\sqrt{x}}\)
Ta có : \(\sqrt{x}+\dfrac{1}{\sqrt{x}}\ge2\sqrt{\sqrt{x}.\dfrac{1}{\sqrt{x}}}\)
=2
Vậy :\(\sqrt{x}+\dfrac{1}{\sqrt{x}}+1\ge2+1\)
Hay\(\sqrt{x}+\dfrac{1}{\sqrt{x}}+1\ge3\)
Min B= 3 Dấu "=" xảy ra khi x=1
CHÚC BẠN HỌC TỐT
\(A=\sum\sqrt{\dfrac{ab}{c+ab}}=\sum\sqrt{\dfrac{ab}{c^2+ca+cb+ab}}\)
\(=\sum\sqrt{\dfrac{ab}{\left(c+a\right)\left(c+b\right)}}\le\dfrac{1}{2}\left(\dfrac{a}{c+a}+\dfrac{b}{c+b}+\dfrac{b}{a+b}+\dfrac{c}{a+c}+\dfrac{a}{b+a}+\dfrac{c}{b+c}\right)\)
\(=\dfrac{1}{2}.3=\dfrac{3}{2}\)
a, \(\sqrt{b}\) tồn tại \(\Leftrightarrow b>0\)
\(\left\{{}\begin{matrix}\sqrt{b}+1\ne0\\\sqrt{b}-1\ne0\\b-1\ne0\end{matrix}\right.\Leftrightarrow b\ne1\)
Vậy B có nghĩa khi \(\left\{{}\begin{matrix}b>0\\b\ne1\end{matrix}\right.\)
b,
\(B=\dfrac{\sqrt{b}}{\sqrt{b}+1}-\dfrac{\sqrt{b}}{\sqrt{b}-1}-\dfrac{2}{b-1}\)
\(=\dfrac{\sqrt{b}}{\sqrt{b}+1}-\dfrac{\sqrt{b}}{\sqrt{b}-1}-\dfrac{2}{\left(\sqrt{b}-1\right)\left(\sqrt{b}+1\right)}\)
\(=\dfrac{\sqrt{b}\left(\sqrt{b}-1\right)-\sqrt{b}\left(\sqrt{b}+1\right)-2}{\left(\sqrt{b}-1\right)\left(\sqrt{b}+1\right)}=\dfrac{b-\sqrt{b}-b-\sqrt{b}-2}{\left(\sqrt{b}-1\right)\left(\sqrt{b}+1\right)}=\dfrac{-2\left(\sqrt{b}+1\right)}{\left(\sqrt{b}-1\right)\left(\sqrt{b}+1\right)}=\dfrac{-2}{\sqrt{b}-1}=\dfrac{2}{1-\sqrt{b}}\)
c,
\(B>1\Leftrightarrow2>1-\sqrt{b}\)
\(\Leftrightarrow2-\left(1-\sqrt{b}\right)=1+\sqrt{b}>0\) (luôn đúng với mọi b)
=> Với mọi b có ĐKXĐ là b khác 0 và b > 1 thì B > 1
\(=9\sqrt{ab}-6\sqrt{ab}+\dfrac{1}{b}\cdot3b\sqrt{ab}\)
\(=3\sqrt{ab}+3\sqrt{ab}=6\sqrt{ab}\)