Lời giải:

Ta có:

\(T=\frac{3+x}{x}+\frac{6-x}{3-x}=\frac{3}{x}+1+\frac{3}{3-x}+1\)

\(=3\left(\frac{1}{x}+\frac{1}{1-x}\right)+2=\frac{9}{x(3-x)}+2\)

Vì \(x\in [1,2]\Rightarrow x,3-x>0\)

Áp dụng BĐT Cauchy ngược dấu: \(x(3-x)\leq \left(\frac{x+3-x}{2}\right)^2=\frac{9}{4}\)

\(\Rightarrow T\geq \frac{9}{\frac{9}{4}}+2=6\) hay \(T_{\min}=6\)

Dấu bằng xảy ra khi \(x=3-x\Leftrightarrow x=\frac{3}{2}\)

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Mặt khác: \(1\leq x\leq 2\Rightarrow (x-1)(x-2)\leq 0\)

\(\Leftrightarrow 3x-x^2\geq 2\Leftrightarrow x(3-x)\geq 2\)

\(\Rightarrow T\leq \frac{9}{2}+2=\frac{13}{2}\)

Vậy \(T_{\max}=\frac{13}{2}\Leftrightarrow \text{x=1 or x=2} \)

Áp dụng bđt : \(\dfrac{1}{a}\)+ \(\dfrac{1}{b}\) ≥ \(\dfrac{4}{a+b}\)(dấu "=" xảy ra ⇔ a=b)

⇒ P= \(\dfrac{1}{x+1}\)+ \(\dfrac{1}{y+2}\) ≥ \(\dfrac{4}{x+1+y+2}\) = \(\dfrac{4}{3+3}\) = \(\dfrac{2}{3}\)

Vậy P_min=\(\dfrac{3}{2}\) ; dấu '=" xảy ra ⇔ \(\left\{{}\begin{matrix}x+1=y+2\\x+y=3\end{matrix}\right.\) ⇔ \(\left\{{}\begin{matrix}x=2\\y=1\end{matrix}\right.\)

Bạn cần nêu rõ ra gt đầu là \(0\le x< 1\) và \(2\leq y<3\) hay là \(0\le x< 1,2=\dfrac{6}{5}\le y< 3\)

\(x+2y=6\)

\(\Leftrightarrow\dfrac{6}{2}=\dfrac{x}{2}+y\)

\(P+\dfrac{6}{2}=\dfrac{8}{x}+\dfrac{1}{y}+\dfrac{x}{2}+y\)

\(\Leftrightarrow P+\dfrac{6}{2}=\left(\dfrac{8}{x}+\dfrac{1}{y}\right)+\left(\dfrac{1}{y}+y\right)\)

vì x;y là số thực dương ,áp dụng BĐT Côsi ta có :

\(\dfrac{8}{x}+\dfrac{x}{2}=2\sqrt{\dfrac{8}{x}+\dfrac{x}{2}}=2\sqrt{4}=2.2=4\)

\(\dfrac{1}{y}+y=2\sqrt{\dfrac{1}{y}+y}=2\sqrt{1}=2.1=2\)

nên \(P+\dfrac{6}{2}\ge6\)

\(\Leftrightarrow P\ge6-\dfrac{6}{2}\)

\(\Leftrightarrow P\ge3\)

vậy \(P_{min}=3\)

\(B=\dfrac{1}{x^3+y^3}+\dfrac{1}{xy\left(x+y\right)}=\dfrac{1}{x^3+y^3}+\dfrac{3}{3xy\left(x+y\right)}\)

\(B\ge\dfrac{\left(1+\sqrt{3}\right)^2}{x^3+y^3+3xy\left(x+y\right)}=\dfrac{4+2\sqrt{3}}{\left(x+y\right)^3}=4+2\sqrt{3}\)

\(B_{min}=4+2\sqrt{3}\) khi \(\left(x;y\right)=\left(\dfrac{3+\sqrt{3}-\sqrt[4]{12}}{6+2\sqrt{3}};\dfrac{3+\sqrt{3}+\sqrt[4]{12}}{6+2\sqrt{3}}\right)\) và hoán vị

Lời giải:

Áp dụng BĐT Cauchy-Shwarz:

$B=\frac{1}{x^3+y^3}+\frac{1}{xy}=\frac{1}{(x+y)^3-3xy(x+y)}+\frac{1}{xy}$

$=\frac{1}{1-3xy}+\frac{1}{xy}=\frac{1}{1-3xy}+\frac{3}{3xy}$

$\geq \frac{(1+\sqrt{3})^2}{1-3xy+3xy}=(1+\sqrt{3})^2$

Vậy $B_{\min}=(1+\sqrt{3})^2$

Dấu "=" xảy ra khi $xy=\frac{1}{2}-\frac{1}{2\sqrt{3}}$

\(x,y,z>0\)

Áp dụng BĐT Caushy cho 3 số ta có:

\(x^3+y^3+z^3\ge3\sqrt[3]{x^3y^3z^3}=3xyz\ge3.1=3\)

\(P=\dfrac{x^3-1}{x^2+y+z}+\dfrac{y^3-1}{x+y^2+z}+\dfrac{z^3-1}{x+y+z^2}\)

\(=\dfrac{\left(x^3-1\right)^2}{\left(x^2+y+z\right)\left(x^3-1\right)}+\dfrac{\left(y^3-1\right)^2}{\left(x+y^2+z\right)\left(y^3-1\right)}+\dfrac{\left(z^3-1\right)^2}{\left(x+y+z^2\right)\left(x^3-1\right)}\)

Áp dụng BĐT Caushy-Schwarz ta có:

\(P\ge\dfrac{\left(x^3+y^3+z^3-3\right)^2}{\left(x^2+y+z\right)\left(x^3-1\right)+\left(x+y^2+z\right)\left(y^3-1\right)+\left(x+y^2+z\right)\left(y^3-1\right)}\)

\(\ge\dfrac{\left(3-3\right)^2}{\left(x^2+y+z\right)\left(x^3-1\right)+\left(x+y^2+z\right)\left(y^3-1\right)+\left(x+y^2+z\right)\left(y^3-1\right)}=0\)

\(P=0\Leftrightarrow x=y=z=1\)

Vậy \(P_{min}=0\)

bạn có thể xem lại điều kiện của x+y+z đc k ạ

\(T=\frac{3+x}{x}+\frac{6-x}{3-x}=\frac{\left(3+x\right)\left(3-x\right)+x\left(6-x\right)}{x\left(3-x\right)}=\frac{9-x^2+6x-x^2}{x\left(3-x\right)}=\frac{9+6x-2x^2}{x\left(3-x\right)}\)

Đặt T = a

<=> \(\frac{9+6x-2x^2}{x\left(3-x\right)}=a\)

<=> \(9+6x-2x^2=3xa-x^2a\)

<=> \(2x^2-6x-9=x^2a-3xa\)

<=> \(x^2\left(2-a\right)-x\left(6-3a\right)-9=0\)

Phương trình trên có nghiệm 

<=> \(\Delta=\left(6-3a\right)^2+4.9.\left(2-a\right)\ge0\)

<=> \(36-36a+9a^2+72-36a\ge0\)

<=> \(9a^2-72a+108\ge0\)

<=> \(\left(a-6\right)\left(a-2\right)\ge0\)

<=> \(\hept{\begin{cases}a\ge6\\a\le2\end{cases}}\)

Vậy \(Min_T=6\) <=> \(x=\frac{3}{2}\)

và \(Max_T=2\Leftrightarrow x\in\varnothing\) (Không tồn tại giá trị lớn nhất của x )