R=\(x\sqrt{3-x^2}=\sqrt{x^2\left(3-x^2\right)}\)
AD BĐT cô si \(a+b\ge2\sqrt{ab}\)
=>\(R=\sqrt{x^2\left(3-x^2\right)}\le\dfrac{x^2+3-x^2}{2}=\dfrac{3}{2}\)
Vậy GTLN của R=\(\dfrac{3}{2}\Leftrightarrow x^2=3-x^2\Leftrightarrow x=\dfrac{\sqrt{6}}{2}\)

\(R=x\sqrt{3-x^2}=\sqrt{x^2\left(3-x^2\right)}\le\dfrac{x^2+3-x^2}{2}=\dfrac{3}{2}\)

\(\Rightarrow R_{max}=\dfrac{3}{2}\) khi \(x^2=3-x^2\Rightarrow x=\dfrac{\sqrt{6}}{2}\)

Lời giải:
$M(2\sqrt{x}-3)=\sqrt{x}+2$

$\Leftrightarrow \sqrt{x}(2M-1)=3M-2$

$\Leftrightarrow x=(\frac{3M-2}{2M-1})^2$

Vì $x$ nguyên nên $\frac{3M-2}{2M-1}$ nguyên 

$\Rightarrow 3M-2\vdots 2M-1$

$\Leftrightarrow 6M-4\vdots 2M-1$
$\Leftrightarrow 3(2M-1)-1\vdots 2M-1$
$\Leftrightarrow 1\vdots 2M-1$

$\Rightarrow 2M-1\in\left\{\pm 1\right\}$

$\Rightarrow M=0;1$

$\Leftrightarrow x=4; 1$ (đều tm)

a: \(A=\dfrac{x-3\sqrt{x}+2x+6\sqrt{x}-3x-9}{x-9}=\dfrac{-3\sqrt{x}-9}{x-9}\)

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

b: A=1/3

=>\(\dfrac{-3}{\sqrt{x}-3}=\dfrac{1}{3}\)

=>căn x-3=-9

=>căn x=-6(loại)

c: căn x-3>=-3

=>3/căn x-3<=-1

=>-3/căn x-3>=1

Dấu = xảy ra khi x=0

Bạn tham khảo ở câu hỏi này :

Câu hỏi của Vampire - Toán lớp 9 | Học trực tuyến

1:

ĐKXĐ: x≠4

Ta có: \(x=\sqrt{7-4\sqrt{3}}+\sqrt{7+4\sqrt{3}}\)

\(=\sqrt{3-2\cdot\sqrt{3}\cdot2+4}+\sqrt{3+2\cdot\sqrt{3}\cdot2+4}\)

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

\(=\left|\sqrt{3}-2\right|+\left|\sqrt{3}+2\right|\)

\(=2-\sqrt{3}+\sqrt{3}+2\)

\(=4\)(ktm ĐKXĐ)

Vậy: Khi x=4 thì A không có giá trị

2: Ta có: P=A+B

\(\Leftrightarrow P=\frac{2}{\sqrt{x}-2}+\frac{\sqrt{x}}{x+1}-\frac{4\sqrt{x}+2}{x\sqrt{x}-2x+\sqrt{x}-2}\)

\(\Leftrightarrow P=\frac{2\left(x+1\right)}{\left(\sqrt{x}-2\right)\left(x+1\right)}+\frac{\sqrt{x}\left(\sqrt{x}-2\right)}{\left(x+1\right)\left(\sqrt{x}-2\right)}-\frac{4\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(x+1\right)}\)

\(\Leftrightarrow P=\frac{2x+2+x-2\sqrt{x}-4\sqrt{x}-2}{\left(x+1\right)\left(\sqrt{x}-2\right)}\)

\(\Leftrightarrow P=\frac{3x-6\sqrt{x}}{\left(x+1\right)\left(\sqrt{x}-2\right)}=\frac{3\sqrt{x}\left(\sqrt{x}-2\right)}{\left(x+1\right)\left(\sqrt{x}-2\right)}\)

\(\Leftrightarrow P=\frac{3\sqrt{x}}{x+1}\)

a: Ta có: \(x^2=3-2\sqrt{2}\)

nên \(x=\sqrt{2}-1\)

Thay \(x=\sqrt{2}-1\) vào A, ta được:

\(A=\dfrac{\left(\sqrt{2}+1\right)^2}{\sqrt{2}-1}=\dfrac{3+2\sqrt{2}}{\sqrt{2}-1}=7+5\sqrt{2}\)