≥ 9(a + b + c)

(a2+b2+c2)3(a2+b2+c2)3 ≥ 9(a + b + c)

chúc bn hok tốt

Lời giải:
Áp dụng BĐT Bunhiacopxky:

$(a^2+b^2+c^2)(1+1+1)\geq (a+b+c)^2$

$\Rightarrow a^2+b^2+c^2\geq \frac{(a+b+c)^2}{3}$

$\Rightarrow (a^2+b^2+c^2)^3\geq \frac{(a+b+c)^6}{27}$

Áp dụng BĐT Cô-si: $a+b+c\geq 3\sqrt[3]{abc}=3$

$\Rightarrow (a^2+b^2+c^2)^3\geq \frac{(a+b+c)^6}{27}\geq \frac{(a+b+c).3^5}{27}=9(a+b+c)$
Ta có đpcm

Dấu "=" xảy ra khi $a=b=c=1$

Đặt \(a=\dfrac{yz}{x^2};b=\dfrac{zx}{y^2};c=\dfrac{xy}{z^2}\)

Áp dụng BĐT BSC:

\(\dfrac{1}{a^2+a+1}+\dfrac{1}{b^2+b+1}+\dfrac{1}{c^2+c+1}\)

\(=\dfrac{x^4}{x^4+x^2yz+y^2z^2}+\dfrac{y^4}{y^4+y^2zx+z^2x^2}+\dfrac{z^4}{z^4+z^2xy+x^2y^2}\)

\(\ge\dfrac{\left(x^2+y^2+z^2\right)^2}{x^4+y^4+z^4+x^2y^2+y^2z^2+z^2x^2+xyz\left(x+y+z\right)}\)

Ta cần chứng minh: 

\(\dfrac{\left(x^2+y^2+z^2\right)^2}{x^4+y^4+z^4+x^2y^2+y^2z^2+z^2x^2+xyz\left(x+y+z\right)}\ge1\)

\(\Leftrightarrow\left(x^2+y^2+z^2\right)^2\ge x^4+y^4+z^4+x^2y^2+y^2z^2+z^2x^2+xyz\left(x+y+z\right)\)

\(\Leftrightarrow x^2y^2+y^2z^2+z^2x^2-xy.yz-yz.zx-zx.xy\ge0\)

\(\Leftrightarrow\left(xy-yz\right)^2+\left(yz-zx\right)^2+\left(zx-xy\right)^2\ge0,\forall x,y,z\)

\(\Rightarrow dpcm\)

Đẳng thức xảy ra khi \(a=b=c=1\)

Áp dụng bđt Schwarz ta có:

\(P=\dfrac{a^4}{2ab+3ac}+\dfrac{b^4}{2cb+3ab}+\dfrac{c^4}{2ac+3bc}\ge\dfrac{\left(a^2+b^2+c^2\right)^2}{5\left(ab+bc+ca\right)}\ge\dfrac{\left(a^2+b^2+c^2\right)^2}{5\left(a^2+b^2+c^2\right)}=\dfrac{1}{5}\).

Đẳng thức xảy ra khi và chỉ khi \(a=b=c=\dfrac{\sqrt{3}}{3}\).

\(Áp\ dụng\ BĐT\ AM - GM,\ ta\ có: \\\sum\dfrac{1}{a^2+2b^2+3}=\sum\dfrac{1}{(a^2+b^2)+(b^2+1)+2}\le\sum\dfrac{1}{2ab+2b+2} \\=\dfrac{1}{2}\sum\dfrac{1}{ab+b+1}=\dfrac{1}{2}.1=\dfrac{1}{2} \\Đẳng\ thức\ xảy\ ra\ khi\ a=b=c=1.\)

Cần c/m: \(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\ge3\sqrt{2}\)

Mặt khác \(\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)\left(\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{b+c}}+\frac{1}{\sqrt{c+a}}\right)\ge9\)

Nên ta chỉ cần c/m  \(P=\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{b+c}}+\frac{1}{\sqrt{c+a}}\le\frac{9}{3\sqrt{2}}=\frac{3\sqrt{2}}{2}\)

Ta có

\(P.\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{\left(a+b\right).2}}+\frac{1}{\sqrt{\left(b+c\right).2}}+\frac{1}{\sqrt{\left(c+a\right).2}}\)

\(=\sqrt{\frac{1}{a+b}}.\sqrt{\frac{1}{2}}+\sqrt{\frac{1}{b+c}}.\sqrt{\frac{1}{2}}+\sqrt{\frac{1}{c+a}}.\sqrt{\frac{1}{2}}\)

\(\le\frac{1}{2}\left(\frac{1}{a+b}+\frac{1}{2}\right)+\frac{1}{2}\left(\frac{1}{b+c}+\frac{1}{2}\right)+\frac{1}{2}\left(\frac{1}{c+a}+\frac{1}{2}\right)\)

\(=\frac{1}{2}\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)+\frac{3}{4}\le\frac{1}{8}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}+\frac{1}{c}+\frac{1}{c}+\frac{1}{a}\right)+\frac{3}{4}\)

\(=\frac{1}{4}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+\frac{3}{4}=\frac{1}{4}.3+\frac{3}{4}=\frac{3}{2}\)

Suy ra  \(P\le\frac{3}{2}:\frac{1}{\sqrt{2}}=\frac{3\sqrt{2}}{2}\)

BĐT được c/m

Đẳng thức xảy ra  \(\Leftrightarrow a=b=c=1\)

1 . 

Từ gt : \(2ab+6bc+2ac=7abc\)và \(a,b,c>0\)

Chia cả hai vế cho abc > 0 

\(\Rightarrow\frac{2}{c}+\frac{6}{a}+\frac{2}{b}=7\)

Đặt \(x=\frac{1}{a},y=\frac{1}{b},z=\frac{1}{c}\Rightarrow\hept{\begin{cases}x,y,z>0\\2z+6x+2y=7\end{cases}}\)

Khi đó : \(C=\frac{4ab}{a+2b}+\frac{9ac}{a+4c}+\frac{4bc}{b+c}\)

\(=\frac{4}{2x+y}+\frac{9}{4x+z}+\frac{4}{y+z}\)

\(\Rightarrow C=\frac{4}{2x+y}+2x+y+\frac{9}{4x+z}+4x+z+\frac{4}{y+z}+y+z\)\(-\left(2x+y+4x+z+y+z\right)\)

\(=\left(\frac{2}{\sqrt{x+2y}}-\sqrt{x+2y}\right)^2+\left(\frac{3}{\sqrt{4x+z}}-\sqrt{4x+z}\right)^2\)\(+\left(\frac{2}{\sqrt{y+z}}-\sqrt{y+z}\right)^2+17\ge17\)

Khi \(x=\frac{1}{2},y=z=1\)thì \(C=17\)

Vậy GTNN của C là 17 khi a =2; b =1; c = 1

2 . 

Áp dụng bất đẳng thức Cauchy ta có :\(1+b^2\ge2b\)nên 

\(\frac{a+1}{1+b^2}=\left(a+1\right)-\frac{b^2\left(a+1\right)}{b^2+1}\)

\(\ge\left(a+1\right)-\frac{b^2\left(a+1\right)}{2b}=a+1-\frac{ab+b}{2}\)

\(\Leftrightarrow\frac{a+1}{1+b^2}\ge a+1-\frac{ab+b}{2}\left(1\right)\)

Tương tự ta có:

\(\frac{b+1}{1+c^2}\ge b+1-\frac{bc+c}{2}\left(2\right)\)

\(\frac{c+1}{1+a^2}\ge c+1-\frac{ca+a}{2}\left(3\right)\)

Cộng vế theo vế (1), (2) và (3) ta được: 

\(\frac{a+1}{1+b^2}+\frac{b+1}{1+c^2}+\frac{c+1}{1+a^2}\ge3+\frac{a+b+c-ab-bc-ca}{2}\left(^∗\right)\)

Mặt khác : \(3\left(ab+bc+ca\right)\le\left(a+b+c\right)^2=9\)

\(\Rightarrow\frac{a+b+c-ab-bc-ca}{2}\ge0\)

Nên \(\left(^∗\right)\) \(\Leftrightarrow\frac{a+1}{1+b^2}+\frac{b+1}{1+c^2}+\frac{c+1}{1+a^2}\ge3\left(đpcm\right)\)

Dấu " = " xảy ra khi và chỉ khi \(a=b=c=1\)

Chúc bạn học tốt !!!