1.  x≥1 <=> \(\frac{1}{x}\le1\Leftrightarrow\frac{1}{x}+1\le2\Leftrightarrow A\le2\Rightarrow MaxA=2\Leftrightarrow x=1\)

2. Áp dụng bđt cosi cho x>0. ta có: \(x+\frac{1}{x}\ge2\sqrt{x.\frac{1}{x}}=2\Leftrightarrow P\ge2\Rightarrow MinP=2\Leftrightarrow x=\frac{1}{x}\Leftrightarrow x=1\)

3: \(A=\frac{x^2+x+4}{x+1}=\frac{\left(x^2+2x+1\right)-\left(x+1\right)+4}{x+1}=x+1-1+\frac{4}{x+1}\)

áp dụng cosi cho 2 số dương ta có: \(x+1+\frac{4}{x+1}\ge2\sqrt{x+1.\frac{4}{x+1}}=2\Leftrightarrow A+1\ge2\Rightarrow A\ge3\Rightarrow MinA=3\Leftrightarrow x+1=\frac{4}{x+1}\Leftrightarrow x=1\)

x=1 nhe nhap minh di ma ket ban voi minh nhe

\(a,\left(\frac{x^2-1}{x^4-x^2+1}-\frac{1}{x^2+1}\right)\left(x^4+\frac{1-x^4}{1+x^2}\right)\)

\(=\frac{\left(x^2-1\right)\left(x^2+1\right)-x^4+x^2-1}{\left(x^4-x^2+1\right)\left(x^2+1\right)}\left(x^4+1-x^2\right)\)

\(=\frac{x^4-1-x^4+x^2-1}{x^2+1}\)

\(=\frac{x^2+2}{x^2+1}\)

b, biển đổi \(M=1-\frac{3}{x^2+1}\)

M bé nhất khi \(\frac{3}{x^2+1}\)lớn nhất

\(\Leftrightarrow x^2+1\)bé nhất \(\Leftrightarrow x^2=0\)

\(\Rightarrow x=0\Rightarrow\)M bé nhất =-2

a, \(P=\left(\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{\sqrt{x}}{\sqrt{x}-1}\right):\left(\frac{2}{x}-\frac{2-x}{x\left(\sqrt{x}+1\right)}\right)\)

\(=\left(\frac{2\sqrt{x}}{\sqrt{x}-1}\right):\left(\frac{2\sqrt{x}+2-2+x}{x\left(\sqrt{x}+1\right)}\right)\)

\(=\frac{2\sqrt{x}}{\sqrt{x}-1}:\left(\frac{x+2\sqrt{x}}{x\left(\sqrt{x}+1\right)}\right)=\frac{2x\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}=\frac{2x\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}\)

b, Ta có \(P=4\Rightarrow\frac{2x\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}=4\)

\(\Leftrightarrow2x\left(\sqrt{x}+1\right)=4\left(x+\sqrt{x}-2\right)\)

\(\Leftrightarrow2x\sqrt{x}+2x=4x+4\sqrt{x}-8\Leftrightarrow2x\sqrt{x}-2x-4\sqrt{x}+8=0\)

Ps : bạn kiểm tra lại đề nhé, nhìn phần a thôi thấy sai rồi 

a.ĐKXĐ \(x\ne0,x\ne1\),\(x\ne-1\)

B=\(\frac{4}{\left(x-1\right)^2}-\frac{x^2-1}{x^3-x}.\frac{x^3+x}{\left(x-1\right)^2}\)=\(\frac{4}{\left(x-1\right)^2}-\frac{x.\left(x^2+1\right)\left(x^2-1\right)}{x\left(x^2-1\right)\left(x-1\right)^2}\)=\(\frac{4}{\left(x-1\right)^2}-\frac{x^2+1}{\left(x-1\right)^2}\)

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

b.TH1 x=3\(\Rightarrow\)B=\(\frac{3-3^2}{2^2}=\frac{-3}{2}\)

TH2 x=-1\(\Rightarrow\)B=\(\frac{3-\left(-1\right)^2}{4}=\frac{1}{2}\)

c.B=-1\(\Leftrightarrow\frac{3-x^2}{\left(x-1\right)^2}=-1\)\(\Leftrightarrow x^2-3=x^2-2x+1\)\(\Leftrightarrow2x=4\Leftrightarrow x=2\)

d.B+2=\(\frac{3-x^2}{\left(x-1\right)^2}+2=\frac{x^2-4x+5}{\left(x-1\right)^2}=\frac{\left(x-2\right)^2+1}{\left(x-1\right)^2}\ge0\)với mọi x\(\Rightarrow B\)>-2

\(a;b>0\)
\(a+b\ge2\sqrt{ab};\frac{1}{a}+\frac{1}{b}\ge2\sqrt{\frac{1}{ab}}\)
\(\Rightarrow\left(a+b\right)\left(\frac{1}{a}+\frac{1}{b}\right)\ge4\Rightarrow\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\)
Dấu "=" xảy ra <=> a=b

a/d bđt \(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\)
\(\Rightarrow P\ge\frac{4}{9-x+x+4}=\frac{4}{13}\)
Dấu "=" xảy ra <=>\(9-x=x+4\)<=>\(x=\frac{5}{2}\)

1. Ta có : \(A=\frac{\left(x+4\right)\left(x+9\right)}{x}=\frac{x^2+13x+36}{x}=x+\frac{36}{x}+13\)

Áp dụng bđt Cauchy : \(x+\frac{36}{x}\ge2\sqrt{x.\frac{36}{x}}=12\)

\(\Rightarrow A\ge25\)

Vậy Min A = 25 \(\Leftrightarrow\begin{cases}x>0\\x=\frac{36}{x}\end{cases}\) \(\Leftrightarrow x=6\)

2. \(B=\frac{\left(x+100\right)^2}{x}=\frac{x^2+200x+100^2}{x}=x+\frac{100^2}{x}+200\)

Áp dụng bđt Cauchy : \(x+\frac{100^2}{x}\ge2\sqrt{x.\frac{100^2}{x}}=200\)

\(\Rightarrow B\ge400\)

Vậy Min B = 400 \(\Leftrightarrow\begin{cases}x>0\\x=\frac{100^2}{x}\end{cases}\) \(\Leftrightarrow x=100\)

Lời giải:
1. Áp dụng BĐT Cô-si

$G=\frac{x^2}{x-1}=\frac{(x^2-1)+1}{x-1}=x+1+\frac{1}{x-1}$

$=(x-1)+\frac{1}{x-1}+2$
$\geq 2\sqrt{(x-1).\frac{1}{x-1}}+2=2+2=4$ 

Vậy $G_{\min}=4$. Giá trị này đạt tại $x-1=\frac{1}{x-1}$

$\Leftrightarrow x=0$ hoặc $x=2$

2.

Áp dụng BĐT Cô-si:

$H=x+\frac{1}{x}=(\frac{x}{4}+\frac{1}{x})+\frac{3}{4}x$

$\geq 2\sqrt{\frac{x}{4}.\frac{1}{x}}+\frac{3}{4}x$
$=1+\frac{3}{4}x\geq 1+\frac{3}{4}.2=\frac{5}{2}$ (do $x\geq 2$)

Vậy $H_{\min}=\frac{5}{2}$. Giá trị này đạt tại $x=2$
 

3.

Áp dụng BĐT Cô-si:

$K=x^2+\frac{1}{x}=(\frac{x^2}{54}+\frac{1}{2x}+\frac{1}{2x})+\frac{53}{54}x^2$

$\geq 3\sqrt[3]{\frac{x^2}{54}.\frac{1}{2x}.\frac{1}{2x}}+\frac{53}{54}x^2$
$=\frac{1}{2}+\frac{53}{54}x^2\geq \frac{1}{2}+\frac{53}{54}.3^2=\frac{28}{3}$ (do $x\geq 3$)

Vậy $K_{\min}=\frac{28}{3}$ khi $x=3$