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Ta có :
\(a^2b+b^2c+c^2a\ge\frac{9a^2b^2c^2}{1+2a^2b^2c^2}\)
\(\Leftrightarrow\left(a^2b+b^2c+c^2a\right)\left(1+2a^2b^2c^2\right)\ge9a^2b^2c^2\)
\(\Leftrightarrow a^2b+b^2c+c^2a+2a^4b^3c^2+2a^2b^4c^{3v}+2a^3b^2c^4\ge3a^2b^2c^2\left(a+b+c\right)\)(*)
Áp dụng BĐT AM-GM ta có:
\(a^2b+a^4b^3c^2+a^3b^2c^4\ge3\sqrt[3]{a^9b^6c^6}=3a^3b^2c^2\)
\(b^2c+a^2b^4c^3+a^4b^3c^2\ge3a^2b^3c^2\)
\(c^2a+a^3b^2c^4+a^2b^4c^4\ge3a^2b^2c^3\)
Cộng theo vế
\(\Rightarrow a^2b+b^2c+c^2a+2a^4b^3c^2+2a^2b^4c^3+2a^3b^2c^4\ge3a^2b^2c^2\left(a+b+c\right)\)
Vậy $(*)$ đúng
Do đó ta có đpcm
#Cừu
![](https://rs.olm.vn/images/avt/0.png?1311)
a/ Nếu (a + b) < 0 thì bất đẳng thức đúng
Với (a + b) \(\ge0\)thì ta có
\(2a^2+ab+2b^2\ge\frac{5}{4}\left(a^2+2ab+b^2\right)\)
\(\Leftrightarrow3a^2-6ab+3b^2\ge0\)
\(\Leftrightarrow3\left(a-b\right)^2\ge0\)(đúng)
b/ Áp dụng BĐT BCS :
\(1=\left(1.\sqrt{a}+1.\sqrt{b}+1.\sqrt{c}\right)^2\le3\left(a+b+c\right)\Rightarrow a+b+c\ge\frac{1}{3}\)
Áp dụng câu a/ :
\(\sqrt{2a^2+ab+2b^2}\ge\frac{\sqrt{5}}{2}\left(a+b\right)\)
\(\sqrt{2b^2+bc+2c^2}\ge\frac{\sqrt{5}}{2}\left(b+c\right)\)
\(\sqrt{2c^2+ac+2a^2}\ge\frac{\sqrt{5}}{2}\left(a+c\right)\)
\(\Rightarrow P\ge\frac{\sqrt{5}}{2}.2\left(a+b+c\right)\ge\frac{\sqrt{5}}{3}\)
Đẳng thức xảy ra khi \(a=b=c=\frac{1}{9}\)
Vậy min P = \(\frac{\sqrt{5}}{3}\) khi a=b=c=1/9
![](https://rs.olm.vn/images/avt/0.png?1311)
Đề bài sai
Phản ví dụ: \(a=\dfrac{1}{2};b=2;c=4\) vì VT<VP
![](https://rs.olm.vn/images/avt/0.png?1311)
Lời giải:
Ta có: \(a^2b+b^2c+c^2a\geq \frac{9a^2b^2c^2}{1+2a^2b^2c^2}\)
\(\Leftrightarrow (a^2b+b^2c+c^2a)(1+2a^2b^2c^2)\geq 9a^2b^2c^2\)
\(\Leftrightarrow a^2b+b^2c+c^2a+2a^4b^3c^2+2a^2b^4c^3+2a^3b^2c^4\geq 3a^2b^2c^2(a+b+c)(*)\)
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Áp dụng BĐT AM-GM ta có:
\(a^2b+a^4b^3c^2+a^3b^2c^4\geq 3\sqrt[3]{a^9b^6c^6}=3a^3b^2c^2\)
\(b^2c+a^2b^4c^3+a^4b^3c^2\geq 3a^2b^3c^2\)
\(c^2a+a^3b^2c^4+a^2b^4c^3\geq 3a^2b^2c^3\)
Cộng theo vế:
\(\Rightarrow a^2b+b^2c+c^2a+2a^4b^3c^2+2a^2b^4c^3+2a^3b^2c^4\geq 3a^2b^2c^2(a+b+c)\)
Vậy $(*)$ đúng
Do đó ta có đpcm
Dấu bằng xảy ra khi $a=b=c=1$
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\dfrac{x-4+5}{\sqrt{x}-2}=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)+5}{\sqrt{x}-2}=\sqrt{x}+2+\dfrac{5}{\sqrt{x}-2}\)
\(=\sqrt{x}-2+\dfrac{5}{\sqrt{x}-2}+4\ge2\sqrt{\dfrac{5\left(\sqrt{x}-2\right)}{\sqrt{x}-2}}+4=4+2\sqrt{5}\)
\(A_{min}=4+2\sqrt{5}\) khi \(9+4\sqrt{5}\)
b.
Đặt \(\left(a;b;c\right)=\left(\dfrac{1}{x};\dfrac{1}{y};\dfrac{l}{z}\right)\Rightarrow xyz=1\)
\(B=\dfrac{x^2}{y+z}+\dfrac{y^2}{z+x}+\dfrac{z^2}{x+y}\ge\dfrac{\left(x+y+z\right)^2}{2\left(x+y+z\right)}=\dfrac{x+y+z}{2}\ge\dfrac{3\sqrt[3]{xyz}}{2}=\dfrac{3}{2}\)
\(B_{min}=\dfrac{3}{2}\) khi \(x=y=z=1\Rightarrow a=b=c=1\)
![](https://rs.olm.vn/images/avt/0.png?1311)
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![](https://rs.olm.vn/images/avt/0.png?1311)
\(P=\dfrac{a^2}{ab+\dfrac{1}{b}}+\dfrac{b^2}{bc+\dfrac{1}{c}}+\dfrac{c^2}{ca+\dfrac{1}{a}}\ge\dfrac{\left(a+b+c\right)^2}{ab+bc+ca+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}}\)
\(P\ge\dfrac{3\left(ab+bc+ca\right)}{ab+bc+ca+\dfrac{ab+bc+ca}{abc}}=\dfrac{3}{1+\dfrac{1}{abc}}=\dfrac{3abc}{1+abc}\) (đpcm)
Dấu "=" xảy ra khi \(a=b=c\)
Với a, b, c > 0 có:
\(P=\dfrac{a}{b+2c}+\dfrac{b}{c+2a}+\dfrac{c}{a+2b}\\ =\dfrac{a^2}{a\left(b+2c\right)}+\dfrac{b^2}{b\left(c+2a\right)}+\dfrac{c^2}{c\left(a+2b\right)}\)
\(\Rightarrow P\ge\dfrac{\left(a+b+c\right)^2}{\left(1+\alpha\right)\left(ab+bc+ca\right)}\ge\dfrac{3\left(ab+bc+ca\right)}{\left(1+\alpha\right)\left(ab+bc+ca\right)}\)
chọn \(\alpha=\dfrac{1}{abc}\Rightarrow dpcm\)
Đề bài : Cho \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\left(a,b,c\ne0\right)\)và \(M=\frac{b^2c^2}{a}+\frac{c^2a^2}{b}+\frac{a^2b^2}{c}\)
Chứng minh M=3abc.
Trước tiên, ta chứng minh bài toán phụ : Cho x+y+z=0 . Chứng minh \(x^3+y^3+z^3=3xyz\)
Giải bài toán phụ như sau : Ta có : \(x+y+z=0\Rightarrow z=-\left(x+y\right)\Rightarrow z^3=-\left[x^3+y^3+3xy\left(x+y\right)\right]\)
\(\Rightarrow x^3+y^3+z^3=-3xy\left(x+y\right)=-3xy\left(-z\right)\)
\(\Rightarrow x^3+y^3+z^3=3xyz\)
Áp dụng vào bài đã cho, ta suy ra : \(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}=\frac{3}{abc}\)
Do đó : \(M=\frac{b^2c^2}{a}+\frac{c^2a^2}{b}+\frac{a^2b^2}{c}=\frac{a^2b^2c^2}{a^3}+\frac{a^2b^2c^2}{b^3}+\frac{a^2b^2c^2}{c^3}=a^2b^2c^2\left(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\right)=a^2b^2c^2.\frac{3}{abc}=3abc\)Vậy \(M=3abc\)(đpcm)
Cảm ơn bạn nha :*