Bài 1: Cho $x,y,z>0$. Tìm Min $P=\frac{2}{x+\sqrt{xy}+\sqrt[3]{xyz}}-\frac{3}{\sqrt{x+y+z}}$

Bài 2: Cho $x,y>0$ thỏa mãn: $x^2y+xy^2=x+y+3xy$. Tìm GTNN của $P=x^2+y^2+\frac{(1+2xy)^2-3}{2xy}$

cho điểm danh ké zới :)) –  Confusion 05-08-16 10:37 PM
2222222222 –  tritanngo99 02-08-16 03:03 PM
hello !!! –  iloveyouproht 02-08-16 02:25 PM
1.
ta có $a+\sqrt{ab}+\sqrt[3]{abc}=a+\sqrt{\frac{1}{2}a.2b} +\sqrt[3]{\frac{1}{4}a.b.4c}$
        $\leq a+\frac{1}{4}a+b+\frac{1}{12}a+ \frac{b}{3}+\frac{4}{3}c =\frac{4}{3}(a+b+c)$
$\Rightarrow P\geq \frac{3}{2(a+b+c)}-\frac{3}{\sqrt{a+b+c}}$ Đặt $t=\sqrt{a+b+c}, t>0$
P $\geq \frac{3}{2t^{2}}-\frac{3}{t}=\frac{3}{2}(\frac{1}{t}-1)^{2}-\frac{3}{2}\geq \frac{-3}{2}$
Dấu '=' $\Leftrightarrow a=\frac{16}{21};b=\frac{4}{21};c=\frac{1}{21}$
2) Có $xy(x+y)=x+y+3xy$
Đặt $\begin{cases}x+y=a \\ xy=b \end{cases}\Rightarrow a^2\geq 4b$
Khi đó : $ab=a+3b\Leftrightarrow 1=\frac{1}{b}+\frac{3}{a}\geq \frac{4}{a^2}+\frac{3}{a}$
$\Rightarrow \frac{1}{a}\leq \frac{1}{4}\Rightarrow a\geq 4$
$P=x^2+y^2+\frac{4x^2y^2+4xy+1-3}{2xy}=(x^2+y^2+2xy)+2-\frac{1}{xy}$
$=(x+y)^2+2-\frac{1}{xy}=a^2+2-\frac{1}{b}=a^2+2+\frac{3}{a}-1$
$=a^2+\frac{3}{a}+1$
Từ đây xét hàm $f(a)=a^2+\frac{3}{a}+1$ trong khoảng $[4 ;+ \infty)$
Ra Min là $17,75 $ khi $a=4$

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