13/4 bn nha

13/4 tick minh nha ban

Bằng 13/4 tick đúng cho mk đi mk chỉ chi tiết cho

a) \(P=\dfrac{3x+3\sqrt{x}-9}{x+\sqrt{x}-2}+\dfrac{\sqrt{x}+3}{\sqrt{x}+2}-\dfrac{\sqrt{x}-2}{\sqrt{x}-1}\left(x\ge0,x\ne1\right)\)

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

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

\(=\dfrac{3x+5\sqrt{x}-8}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}=\dfrac{\left(\sqrt{x}-1\right)\left(3\sqrt{x}+8\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}=\dfrac{3\sqrt{x}+8}{\sqrt{x}+2}\)

b) \(P=\dfrac{3\sqrt{x}+8}{\sqrt{x}+2}=\dfrac{3\sqrt{x}+6+2}{\sqrt{x}+2}=3+\dfrac{2}{\sqrt{x}+2}\)

Để \(P\in Z\Rightarrow2⋮\sqrt{x}+2\Rightarrow\sqrt{x}+2=2\left(\sqrt{x}+2\ge2\right)\)

\(\Rightarrow x=0\)

c) Ta có: \(\sqrt{x}\ge0\Rightarrow\sqrt{x}+2\ge2\Rightarrow\dfrac{2}{\sqrt{x}+2}\le1\Rightarrow3+\dfrac{2}{\sqrt{x}+2}\le4\)

\(\Rightarrow P_{max}=4\) khi \(x=0\)

Đặt \(t=\sqrt{x},t\ge0\)

Dấu "=" xảy ra khi t = 1 <=> x = 1

B đạt giá trị nhỏ nhất bằng 7 tại x = 1

Áp dụng Cô-si:

\(A\le3x+\frac{10-x^2+1}{2}=3x+\frac{11-x^2}{2}\)

\(=\frac{-x^2+6x+11}{2}=\frac{-\left(x^2-6x-11\right)}{2}=\frac{20-\left(x-3\right)^2}{2}\le\frac{20}{2}=10\)

Dấu "=" xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}10-x^2=1\\x-3=0\end{matrix}\right.\)\(\Leftrightarrow x=3\)

@Lê Thị Thục Hiền

@Akai Haruma

\(dk:-\sqrt{2}\le x\le\sqrt{2}\)(*)

\(\left(A-3x\right)=\sqrt{2-x^2}\)

\(\Leftrightarrow a^2-6ax+9x^2=2-x^2\)

\(10x^2-6ax+a^2-2=0\)(**)

Giá trị (a)  (**) có nghiệm thỏa mãn (*)  

(**)\(\Leftrightarrow\left(x^2-2.\frac{3a}{10}x+\frac{9a^2}{100}\right)=\frac{20-a^2}{100}\)\(\Leftrightarrow\left(x-\frac{3a}{10}\right)^2=\frac{20-a^2}{100}\Rightarrow20-a^2\ge0\Rightarrow!a!\le2\sqrt{5}\)

\(a=2\sqrt{5}\Rightarrow x=\frac{6\sqrt{5}}{10}=\frac{3\sqrt{5}}{5}< \sqrt{2}\)(nhạn)

Kết luận: GTLN của A là \(A_{max}=2\sqrt{5}\) tại  \(x=\frac{3\sqrt{5}}{5}\)

\(A=3x+\sqrt{2-x^2}\)

\(\Leftrightarrow\frac{\sqrt{10}A}{2}=\frac{3\sqrt{10}x}{2}+\frac{\sqrt{10}\sqrt{2-x^2}}{2}\)

\(\le\sqrt{\left(\frac{45}{2}+\frac{5}{2}\right)\left(x^2+2-x^2\right)}=\sqrt{25.2}=5\sqrt{2}\)

\(\Rightarrow1A\le\frac{5\sqrt{2}.2}{\sqrt{10}}=2\sqrt{5}\)

Vậy GTLN là A = \(2\sqrt{5}\)khi x = \(\frac{3}{\sqrt{5}}\)  

NguNhuMinh oi

A không có GTLN bạn nhé. A có GTNN thôi.

\(3=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{xy}\Leftrightarrow x+y+1=3xy\)

\(\Leftrightarrow y\left(3x-1\right)=x+1\Leftrightarrow y=\dfrac{x+1}{3x-1}\)

\(\left(3x^2+1\right)\left(3+1\right)\ge\left(3x+1\right)^2\Rightarrow\sqrt{3x^2+1}\ge\dfrac{1}{2}\left(3x+1\right)\)

\(\Rightarrow\dfrac{2}{\sqrt{3x^2+1}}\le\dfrac{4}{3x+1}\)

\(\Rightarrow A\le\dfrac{4}{3x+1}+\dfrac{4}{3y+1}=\dfrac{4}{3x+1}+\dfrac{2\left(3x-1\right)}{3x+1}=\dfrac{6x+2}{3x+1}=2\)

\(A_{min}=2\) khi \(x=y=1\)

Áp dụng bđt bu - nhi -a, ta có 

\(A^2\le\left(3^2+1\right)\left(x^2+2-x\right)=20\Rightarrow A\le2\sqrt{5}\)

dấu = xayra <=>\(\frac{x}{3}=\sqrt{2-x^2}\Leftrightarrow9\left(2-x^2\right)=x^2\Leftrightarrow18=10x^2\Leftrightarrow x=\frac{3}{\sqrt{5}}\)

Bạn ơi, mình chưa hiểu chỗ này lắm, tại sao \(\left(3^2\text{+}1\right)\left(x^2\text{+}2-x\right)\text{=}20\)

chết hơi nhầm chút, phải là (x^2+2-x^2) bạn ạ!Thông Cảm ^_^