cho a,b,c >0.Chứng minh rằng
\(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\ge\dfrac{a+b}{c+a}+\dfrac{b+c}{a+b}+\dfrac{c+a}{b+c}\)
Cho a,b,c >0 Chứng minh rằng:
a) \(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\ge\dfrac{a+b+c}{\sqrt[3]{abc}}\)
b) \(\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ca}{b}\ge\sqrt{3\left(a^2+b^2+c^2\right)}\)
Cho a,b,c>0. Chứng minh rằng: \(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+\dfrac{\sqrt[3]{abc}}{a+b+c}\ge\dfrac{10}{3}\)
Từ bài toán này (mà bạn đã hỏi cách đây vài bữa):
Ta có: \(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\ge\dfrac{a+b+c}{\sqrt[3]{abc}}\)
Do đó: \(VT\ge\dfrac{a+b+c}{\sqrt[3]{abc}}+\dfrac{\sqrt[3]{abc}}{a+b+c}\)
Lại có: \(\dfrac{a+b+c}{\sqrt[3]{abc}}\ge\dfrac{3\sqrt[3]{abc}}{\sqrt[3]{abc}}=3\)
Đặt \(\dfrac{a+b+c}{\sqrt[3]{abc}}=x\ge3\Rightarrow VT\ge x+\dfrac{1}{x}=\dfrac{x}{9}+\dfrac{1}{x}+\dfrac{8x}{9}\ge2\sqrt{\dfrac{x}{9x}}+\dfrac{8}{9}.3=\dfrac{10}{3}\) (đpcm)
cho a,b,c>0. Chứng minh rằng: \(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\ge\dfrac{a+b+c}{\sqrt[3]{abc}}\)
Ta có:
\(\dfrac{a}{b}+\dfrac{a}{b}+\dfrac{b}{c}\ge3\sqrt[3]{\dfrac{a^2}{bc}}=\dfrac{3a}{\sqrt[3]{abc}}\)
\(\dfrac{b}{c}+\dfrac{b}{c}+\dfrac{c}{a}\ge\dfrac{3b}{\sqrt[3]{abc}}\)
\(\dfrac{c}{a}+\dfrac{c}{a}+\dfrac{a}{b}\ge\dfrac{3c}{\sqrt[3]{abc}}\)
Cộng vế:
\(3\left(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\right)\ge\dfrac{3\left(a+b+c\right)}{\sqrt[3]{abc}}\)
\(\Rightarrow\) đpcm
cho a,b,c\(\ge\)0,a+b+c=1.chứng minh rằng
\(\dfrac{a}{1+a^2}+\dfrac{b}{1+b^2}+\dfrac{c}{1+c^2}\le\dfrac{9}{10}\)
Ta có đánh giá sau với a không âm:
\(\dfrac{a}{1+a^2}\le\dfrac{36a+3}{50}\)
Thật vậy, BĐT tương đương:
\(\left(36a+3\right)\left(a^2+1\right)\ge50a\)
\(\Leftrightarrow\left(3a-1\right)^2\left(4a+3\right)\ge0\) (luôn đúng)
Tương tự: \(\dfrac{b}{1+b^2}\le\dfrac{36b+3}{50}\) ; \(\dfrac{c}{1+c^2}\le\dfrac{36c+3}{50}\)
Cộng vế: \(VT\le\dfrac{36\left(a+b+c\right)+9}{50}=\dfrac{9}{10}\)
Dấu "=" xảy ra khi \(a=b=c=\dfrac{1}{3}\)
Ta chứng minh bđt phụ \(\dfrac{a}{1+a^2}\le\dfrac{3}{10}+\dfrac{18}{25}\left(a-\dfrac{1}{3}\right)\)
Thật vậy bđt trên \(\Leftrightarrow\dfrac{-3a^2+10a-3}{10\left(1+a^2\right)}-\dfrac{18}{25}\left(a-\dfrac{1}{3}\right)\le0\)
\(\Leftrightarrow\left(a-\dfrac{1}{3}\right)\left[\dfrac{3\left(3-a\right)}{10\left(1+a^2\right)}-\dfrac{18}{25}\right]\le0\)
\(\Leftrightarrow-\dfrac{36\left(a-\dfrac{1}{3}\right)^2\left(\dfrac{3}{4}+a\right)}{50\left(1+a^2\right)}\le0\) ( luôn đúng với mọi \(a\)\(\ge\)0)
Tương tự cũng có:\(\dfrac{b}{1+b^2}\le\dfrac{3}{10}+\dfrac{18}{25}\left(b-\dfrac{1}{3}\right)\); \(\dfrac{c}{1+c^2}\le\dfrac{3}{10}+\dfrac{18}{25}\left(c-\dfrac{1}{3}\right)\)
Cộng vế với vế => VT\(\le\dfrac{9}{10}+\dfrac{18}{25}\left(a+b+c-1\right)=\dfrac{9}{10}\)
Dấu = xảy ra khi \(a=b=c=\dfrac{1}{3}\)
Cho a, b, c > 0. Chứng minh \(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\ge\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\)
Hình như thế này mới đúng chứ \(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\ge\dfrac{b}{a}+\dfrac{c}{b}+\dfrac{a}{c}\)
Áp dụng BĐT Cosi:
\(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}\ge2.\dfrac{a}{c};\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\ge2.\dfrac{b}{a};\dfrac{c^2}{a^2}+\dfrac{a^2}{b^2}\ge2.\dfrac{c}{b}\)
\(\Rightarrow2\left(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\right)\ge2\left(\dfrac{b}{a}+\dfrac{c}{b}+\dfrac{a}{c}\right)\)
\(\Leftrightarrow\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\ge\dfrac{b}{a}+\dfrac{c}{b}+\dfrac{a}{c}\)
Đẳng thức xảy ra khi \(a=b=c>0\)
Cho a,b,c > 0. Chứng minh rằng
\(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\ge\dfrac{3}{2}\)
Google không tính phí, gõ BĐT Nesbitt là ra
\(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\ge\dfrac{3}{2}\)
\(\Rightarrow\left(\dfrac{a}{b+c}+1\right)+\left(\dfrac{b}{c+a}+1\right)+\left(\dfrac{c}{a+b}+1\right)\ge\dfrac{9}{2}\)
\(\Rightarrow\dfrac{a+b+c}{b+c}+\dfrac{a+b+c}{c+a}+\dfrac{a+b+c}{a+b}\ge\dfrac{9}{2}\)
\(\Rightarrow\left(a+b+c\right)\left(\dfrac{1}{b+c}+\dfrac{1}{c+a}+\dfrac{1}{a+b}\right)\ge\dfrac{9}{2}\)
\(\Rightarrow2\left(a+b+c\right)\left(\dfrac{1}{b+c}+\dfrac{1}{c+a}+\dfrac{1}{a+b}\right)\ge9\)
\(\Rightarrow\left(a+b+c+a+b+c\right)\left(\dfrac{1}{b+c}+\dfrac{1}{c+a}+\dfrac{1}{a+b}\right)\ge9\)
Đặt: \(\left\{{}\begin{matrix}a+b=x\\b+c=y\\c+a=z\end{matrix}\right.\) Khi đó bất đẳng thức trở thành:
\(\left(x+y+z\right)\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\ge9\) (đúng theo AM-GM)
Vậy bất đẳng thức cần chứng minh đúng
Dấu "=" xảy ra khi: \(a=b=c>0\)
Cho a,b,c dương. Chứng minh rằng:
\(\dfrac{a}{b+c}+\dfrac{b+c}{a}+\dfrac{b}{c+a}+\dfrac{c+a}{b}+\dfrac{c}{a+b}+\dfrac{a+b}{c}\ge\dfrac{15}{2}\)
\(A=\dfrac{a}{b+c}+\dfrac{b+c}{a}+\dfrac{b}{c+a}+\dfrac{c+a}{b}+\dfrac{c}{a+b}+\dfrac{a+b}{c}\)
\(A=\left(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\right)+\left(\dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}\right)\)
\(A\ge\dfrac{3}{2}+\dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}\) (bất đẳng thức Nesbit)
\(A\ge\dfrac{3}{2}+\dfrac{b}{a}+\dfrac{c}{a}+\dfrac{c}{b}+\dfrac{a}{b}+\dfrac{a}{c}+\dfrac{b}{c}\)
\(A\ge\dfrac{3}{2}+\left(\dfrac{b}{a}+\dfrac{a}{b}\right)+\left(\dfrac{c}{a}+\dfrac{a}{c}\right)+\left(\dfrac{b}{c}+\dfrac{c}{b}\right)\)
Áp dụng bất đẳng thức AM-GM cho 2 số dương ta có:
\(A\ge\dfrac{3}{2}+2\sqrt{\dfrac{ab}{ab}}+2\sqrt{\dfrac{ac}{ac}}+2\sqrt{\dfrac{bc}{bc}}\)
\(A\ge\dfrac{3}{2}+2+2+2=\dfrac{15}{2}\left(đpcm\right)\)
Dấu"=" xảy ra khi: \(a=b=c\)
Cho a,b,c > 0. Chứng minh rằng :\(\dfrac{a^3}{b}+\dfrac{b^3}{c}+\dfrac{c^3}{a}\ge ab+bc+ac\)
\(VT=\dfrac{a^4}{ab}+\dfrac{b^4}{bc}+\dfrac{c^4}{ca}\ge\dfrac{\left(a^2+b^2+c^2\right)^2}{ab+bc+ca}\ge\dfrac{\left(ab+bc+ca\right)^2}{ab+bc+ca}=ab+bc+ca\)
Dấu "=" xảy ra khi \(a=b=c\)
Ta chứng minh bđt phụ \(x^2+y^2+z^2\ge xy+yz+zx\forall x,y,z>0\)
\(\Leftrightarrow2x^2+2y^2+2z^2\ge2xy+2yz+2zx\Leftrightarrow x^2-2xy+y^2+y^2-2yz+z^2+z^2-2zx+x^2\ge0\Leftrightarrow\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\ge0\)\(\Rightarrow x^2+y^2+z^2\ge xy+yz+zx\left(1\right)\)
Áp dụng bđt Cô-si vào các số a,b,c dương :
\(\dfrac{a^3}{b}+ab\ge2\sqrt{\dfrac{a^3}{b}\cdot ab}=2\sqrt{a^4}=2a^2\)
Chứng minh tương tự ta được:
\(\dfrac{b^3}{c}+bc\ge2b^2;\dfrac{c^3}{a}+ca\ge2c^2\)
\(\Rightarrow\dfrac{a^3}{b}+\dfrac{b^3}{c}+\dfrac{c^3}{a}+ab+bc+ca\ge2a^2+2b^2+2c^2\ge2ab+2bc+2ca\) (do áp dụng (1)) \(\Rightarrow\dfrac{a^3}{b}+\dfrac{b^3}{c}+\dfrac{c^3}{a}\ge2\left(ab+bc+ca\right)-\left(ab+bc+ca\right)=ab+bc+ca\)
Dấu = xảy ra \(\Leftrightarrow a=b=c\)
Cho a,b,c>0.Chứng minh rằng:
\(\dfrac{a+b}{a^2+b^2}+\dfrac{b+c}{b^2+c^2}+\dfrac{a+c}{a^2+c^2}\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
Help me?!