srweafgtseawref

Đặt \(\dfrac{1}{x+1}=a,\dfrac{1}{y+1}=b,\dfrac{1}{z+1}=c\Rightarrow a,b,c>0;a+b+c=1.\)

\(x=\dfrac{1}{a}-1\)

Cần chứng minh: \(\sum\sqrt{\dfrac{1}{a}-1}\le\dfrac{3}{2}\sqrt{\left(\dfrac{1}{a}-1\right)\left(\dfrac{1}{b}-1\right)\left(\dfrac{1}{c}-1\right)}\)

Hay \(\sum\sqrt{\dfrac{1}{a}-\dfrac{1}{a+b+c}}\le\dfrac{3}{2}\sqrt{\prod\left(\dfrac{1}{a}-\dfrac{1}{a+b+c}\right)}\)

Hay là \(\sum\sqrt{\dfrac{b+c}{a\left(a+b+c\right)}}\le\dfrac{3}{2}\sqrt{\prod\dfrac{\left(b+c\right)}{a\left(a+b+c\right)}}\)

Tương đương: \(\sum\sqrt{\dfrac{b+c}{a}}\le\dfrac{3}{2}\sqrt{\prod\dfrac{\left(b+c\right)}{a}}\)

\(\left[\sum\left(b+c\right)\left\{a+2\left(b+c\right)\right\}\right]\left[\sum\dfrac{1}{a\left\{a+2\left(b+c\right)\right\}}\right]\ge\left[\sum\sqrt{\dfrac{b+c}{a}}\right]^2\)

Từ đây cần chứng minh:

\(\dfrac{9}{4}\prod\dfrac{\left(b+c\right)}{a}\ge\left[\sum\left(b+c\right)\left\{a+2\left(b+c\right)\right\}\right]\left[\sum\dfrac{1}{a\left\{a+2\left(b+c\right)\right\}}\right]\)

Còn lại bạn tự làm hoặc không để tối rảnh mình làm.

Do hoc24.vn không cho cập nhật câu trả lời nữa nên mình đăng tiếp:

Thực hiện thay thế \(\left(a,b,c\right)\rightarrow\left(s-a',s-b',s-c'\right)\) với $a',b',c'$ là độ dài ba cạnh của một tam giác.

Đặt $\left\{ \begin{array}{l}a' + b' + c' = 2s\\a'b' + b'c' + c'a' = {s^2} + 4Rr + {r^2}\\a'b'c' = 4sRr\end{array} \right.$

Bất đẳng thức quy về: 

$${\dfrac { \left( 4\,R-24\,r \right) {s}^{4}+r \left( 72\,{R}^{2}+41\,Rr+8\,{r}^{2} \right) {s}^{2}+2\,{r}^{2} \left( 4\,R+r \right) ^{3}}{r{s}^{2} \left( 4\,{s}^{2}+r \left( 8\,R+r \right)  \right) }}\geqslant 0$$

\( \Leftrightarrow \left( {4{\mkern 1mu} R - 24{\mkern 1mu} r} \right){s^4} + r\left( {72{\mkern 1mu} {R^2} + 41{\mkern 1mu} Rr + 8{\mkern 1mu} {r^2}} \right){s^2} + 2{\mkern 1mu} {r^2}{\left( {4{\mkern 1mu} R + r} \right)^3} \geqslant 0\)

Hay là \({s^2}\left( {R - 2{\mkern 1mu} r} \right)\left( {9{\mkern 1mu} {r^2} + 4{\mkern 1mu} {s^2}} \right) + r\left[ {10{\mkern 1mu} {s^2}\left( {4{\mkern 1mu} {R^2} + 4{\mkern 1mu} Rr + 3{\mkern 1mu} {r^2} - {s^2}} \right) + \left( {8{\mkern 1mu} Rr + 2{\mkern 1mu} {r^2} + 2{\mkern 1mu} {s^2}} \right)\left( {16{\mkern 1mu} {R^2} + 8{\mkern 1mu} Rr + {r^2} - 3{\mkern 1mu} {s^2}} \right)} \right] \geqslant 0\)

Đây là điều hiển nhiên.

Ngoài ra phương pháp SOS, SS cũng có thể sử dụng ở đây.

Đặt \(\left(\sqrt{x};\sqrt{y};\sqrt{z}\right)\rightarrow\left(a;b;c\right)\Rightarrow\hept{\begin{cases}a+b+c=1\\a;b;c>0\end{cases}}\)

Và \(\frac{ab}{\sqrt{a^2+b^2+2c^2}}+\frac{bc}{\sqrt{b^2+c^2+2a^2}}+\frac{ca}{\sqrt{c^2+a^2+2b^2}}\le\frac{1}{2}\)

Ta có :

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

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

Tương tự cho 2 BĐT còn lại roouf cộng theo vế :

\(VT\le\frac{1}{2}\left(\frac{ab+bc}{a+c}+\frac{ab+ac}{b+c}+\frac{bc+ac}{a+b}\right)=\frac{1}{2}\left(a+b+c\right)=\frac{1}{2}\)

Dấu " = " xảy ra khi \(a=b=c=\frac{1}{3}\Rightarrow x=y=z=\frac{1}{9}\)

Chúc bạn học tốt !!!

Đặt \(\left(\sqrt{x};\sqrt{y};\sqrt{z}\right)\rightarrow\left(a;b;c\right)\)\(\Rightarrow\left\{{}\begin{matrix}a+b+c=1\\a;b;c>0\end{matrix}\right.\)

Và \(\dfrac{ab}{\sqrt{a^2+b^2+2c^2}}+\dfrac{bc}{\sqrt{b^2+c^2+2a^2}}+\dfrac{ca}{\sqrt{c^2+a^2+2b^2}}\le\dfrac{1}{2}\)

Ta có:\(\dfrac{ab}{\sqrt{a^2+b^2+2c^2}}=\dfrac{2ab}{\sqrt{\left(1+1+2\right)\left(a^2+b^2+2c^2\right)}}\)

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

Tương tự cho 2 BĐT còn lại rồi cộng theo vế:

\(VT\le\dfrac{1}{2}\left(\dfrac{ab+bc}{a+c}+\dfrac{ab+ac}{b+c}+\dfrac{bc+ac}{a+b}\right)\)

\(=\dfrac{1}{2}\left(a+b+c\right)=\dfrac{1}{2}\)

Dấu "=" khi \(a=b=c=\dfrac{1}{3}\Rightarrow x=y=z=\dfrac{1}{9}\)

Bai 1: Ap dung BDT Bunhiacopxki ta co:

         \(ax+by+cz+2\sqrt {(ab+ac+bc)(xy+yz+xz)} \)

         \(≤ \sqrt {(a^2+b^2+c^2)(x^2+y^2+z^2)} + \sqrt {(ab+ac+bc)(xy+yz+zx)}+\sqrt {(ab+ac+bc)(xy+yz+zx)}\)

         \(≤ \sqrt {(a^2+b^2+c^2+2ab+2ac+2bc)(x^2+y^2+z^2+2xy+2yz+2zx)}\)

         \(= (a+b+c)(x+y+z)\) 

   =>  \(Q.E.D\)

Tiep bai 4:Ta co:

               BDT <=>  \((2+y^2z)(2+z^2x)(2+x^2y)≥(2+x)(2+y)(2+z)\)

    Sau khi khai trien con:   \(2(z^2x+y^2z+x^2y)+x^2z+z^2y+y^2x≥xy+yz+zx+2x+2y+2z \)

               Ap dung BDT Cosi ta co:

                                       \(z^2x+x ≥ 2zx \) <=> \(z^2x≥2zx-x\)

              Lam tuong tu ta co:  \(2(z^2x+y^2z+x^2y)≥4xy+4yz+4zx-2x-2y-2z \)(1)

                                        \(x^2z+{1\over z}≥2x \) <=> \(x^2z≥2x-xy \) (do xyz=1)

              Lam tuong tu ta co:  \(x^2z+z^2y+y^2x≥ 2y+2z+2x-xy-yz-zx\)(2)

Cong (1) voi (2) ta co:      VT\(≥ 3(xy+yz+zx)\)(*)

               Voi cach lam tuong tu ta cung duoc:  VT\(≥ 3(x+y+z) \)(**)

Tu (*) va (**) suy ra :   \(3 \)VT \(≥ 6(x+y+z)+3(xy+yz+zx) \)

                           <=>   VT \(≥ 2(x+y+z)+xy+yz+zx\)

                            =>   \(Q.E.D\)

Ta có: \(x+y+z=xyz\Rightarrow x=\frac{x+y+z}{yz}\Rightarrow x^2=\frac{x^2+xy+xz}{yz}\Rightarrow x^2+1=\frac{\left(x+y\right)\left(x+z\right)}{yz}\)\(\Rightarrow\sqrt{x^2+1}=\sqrt{\frac{\left(x+y\right)\left(x+z\right)}{yz}}\le\frac{\frac{x+y}{y}+\frac{x+z}{z}}{2}=1+\frac{x}{2}\left(\frac{1}{y}+\frac{1}{z}\right)\)\(\Rightarrow\frac{1+\sqrt{1+x^2}}{x}\le\frac{2+\frac{x}{2}\left(\frac{1}{y}+\frac{1}{z}\right)}{x}=\frac{2}{x}+\frac{1}{2}\left(\frac{1}{y}+\frac{1}{z}\right)\)

Tương tự: \(\frac{1+\sqrt{1+y^2}}{y}\le\frac{2}{y}+\frac{1}{2}\left(\frac{1}{z}+\frac{1}{x}\right)\); \(\frac{1+\sqrt{1+z^2}}{z}\le\frac{2}{z}+\frac{1}{2}\left(\frac{1}{x}+\frac{1}{y}\right)\)

Cộng theo vế ba bất đẳng thức trên, ta được: \(\frac{1+\sqrt{1+x^2}}{x}+\frac{1+\sqrt{1+y^2}}{y}+\frac{1+\sqrt{1+z^2}}{z}\le3\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=3.\frac{xy+yz+zx}{xyz}\)\(\le3.\frac{\frac{\left(x+y+z\right)^2}{3}}{xyz}=\frac{\left(x+y+z\right)^2}{xyz}=\frac{\left(xyz\right)^2}{xyz}=xyz\)

Đẳng thức xảy ra khi \(x=y=z=\sqrt{3}\)

\(\frac{ax+by+cz}{xy}=z\Rightarrow z=\frac{a}{y}+\frac{b}{x}+\frac{cz}{xy}>\frac{a}{y}+\frac{b}{x}\)

Tương tự có \(y>\frac{a}{z}+\frac{c}{x}\); \(x>\frac{b}{z}+\frac{c}{y}\)

\(\Rightarrow x+y+z>\frac{b+c}{x}+\frac{a+c}{y}+\frac{a+b}{z}=\frac{b+c}{x}+x+\frac{a+c}{y}+y+\frac{a+b}{z}+z-x-y-z\)

\(\Rightarrow2\left(x+y+z\right)>2\sqrt{b+c}+2\sqrt{a+c}+2\sqrt{a+b}\)

\(\Rightarrow x+y+z>\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\)

Từ giả thiết suy ra : \(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}=1\)

Nên ta có : \(\frac{\sqrt{1+x^2}}{x}=\sqrt{\frac{1}{x^2}+\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}}=\sqrt{\left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x}+\frac{1}{z}\right)}\le\frac{1}{2}\left(\frac{2}{x}+\frac{1}{y}+\frac{1}{z}\right)\)

Dấu " = " \(\Leftrightarrow y=z\)

Vậy \(\frac{1+\sqrt{1+x^2}}{x}\le\frac{1}{2}\left(\frac{4}{x}+\frac{1}{y}+\frac{1}{z}\right)\)

Tương tự ta có :

\(\frac{1+\sqrt{1+y^2}}{y}\le\frac{1}{2}\left(\frac{1}{x}+\frac{4}{y}+\frac{1}{z}\right);\frac{1+\sqrt{1+z^2}}{z}\le\frac{1}{2}\left(\frac{1}{x}+\frac{1}{y}+\frac{4}{z}\right)\)

Vậy ta có :

\(\frac{1+\sqrt{1+x^2}}{x}+\frac{1+\sqrt{1+y^2}}{y}+\frac{1+\sqrt{1+z^2}}{z}\le3\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\)

Dấu " = " \(\Leftrightarrow x=y=z\)

Ta có :

\(\left(x+y+z\right)^2-3\left(xy+yz+xx\right)=...=\frac{1}{2}\left[\left(x-y\right)^2+\left(y-z\right)^2+\left(x-z\right)^2\right]\ge0\)

Nên \(\left(x+y+x\right)^2\ge3\left(xy+yz+xx\right)\)

\(\Rightarrow\left(xyz\right)^2\ge3\left(xy+yz+xz\right)\Rightarrow3\frac{xy+yz+xz}{xyz}\Rightarrow3\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\le xyz\)

Vậy \(\frac{1+\sqrt{1+x^2}}{x}+\frac{1+\sqrt{1+y^2}}{y}+\frac{1+\sqrt{1+z^2}}{z}\le xyz\)

Dấu " = " \(\Leftrightarrow x=y=z\)

Chúc bạn học tốt !!

\(\frac{1+\frac{1}{2}.2.\sqrt{1+x^2}}{x}\le\frac{1+\frac{1}{4}\left(x^2+5\right)}{x}=\frac{x}{4}+\frac{9}{4x}\)

\(\Rightarrow VT\le\frac{1}{4}\left(x+y+z\right)+\frac{9}{4}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\)

\(VT\le\frac{1}{4}\left(x+y+z\right)+\frac{9\left(xy+yz+zx\right)}{4xyz}=\frac{1}{4}\left(x+y+z\right)+\frac{9\left(xy+yz+zx\right)}{4\left(x+y+z\right)}\)

\(VT\le\frac{1}{4}\left(x+y+z\right)+\frac{3\left(x+y+z\right)^2}{4\left(x+y+z\right)}=x+y+z=xyz\)

Dấu "=" xảy ra khi \(x=y=z=\sqrt{3}\)