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

Điều kiện: \(x\ge0\) , \(\) \(x\) khác \(1\)

Đổi dấu ở phân thức thứ 3: \(-\frac{1}{1 - \sqrt{x}} = +\frac{1}{\sqrt{x} - 1}\)

Phân tích mẫu thức thứ nhất thành nhân tử:

\(x\sqrt{x} - \sqrt{x} = \sqrt{x}(x - 1) = \sqrt{x}(\sqrt{x} - 1)(\sqrt{x} + 1)\)

Mẫu thức chung trong ngoặc là: \(\sqrt{x}(\sqrt{x}-1)(x+\sqrt{x}+1)=\sqrt{x}(x\sqrt{x}-1)\)

Quy đồng và cộng các phân thức trong ngoặc:

\(\frac{x+2}{\sqrt{x}(\sqrt{x}-1)(\sqrt{x}+1)}\quad\)

\(A = \frac{x+2 + \sqrt{x}\cdot \sqrt{x}(\sqrt{x}-1) + \sqrt{x}(x+\sqrt{x}+1)}{\sqrt{x}(\sqrt{x}-1)(x+\sqrt{x}+1)}\)

\(A = \frac{x+2 + x(\sqrt{x}-1) + x\sqrt{x} + x + \sqrt{x}}{\sqrt{x}(\sqrt{x}-1)(x+\sqrt{x}+1)}\)

\(A = \frac{x + 2 + x\sqrt{x} - x + x\sqrt{x} + x + \sqrt{x}}{\sqrt{x}(\sqrt{x}-1)(x+\sqrt{x}+1)}\)

\(A = \frac{2x\sqrt{x} + x + \sqrt{x} + 2}{\sqrt{x}(\sqrt{x}-1)(x+\sqrt{x}+1)}\)

Phân tích tử số thành nhân tử:

\(2x\sqrt{x} + x + \sqrt{x} + 2 = (2x\sqrt{x} + 2) + (x + \sqrt{x})\)

\(= 2(x\sqrt{x} + 1) + \sqrt{x}(\sqrt{x} + 1)\)

\(= 2(\sqrt{x} + 1)(x - \sqrt{x} + 1) + \sqrt{x}(\sqrt{x} + 1)\)

\(= (\sqrt{x} + 1)\left[2(x - \sqrt{x} + 1) + \sqrt{x}\right]\)

\(= (\sqrt{x} + 1)(2x - \sqrt{x} + 2)\)

Thay vào biểu thức ta được:

\(Q = \frac{(\sqrt{x} + 1)(2x - \sqrt{x} + 2)}{\sqrt{x}(\sqrt{x}-1)(x+\sqrt{x}+1)} \cdot \frac{2}{\sqrt{x}-1}\)

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

\(\Leftrightarrow Q=\frac{2(\sqrt{x} + 1)(2x - \sqrt{x} + 2)}{\sqrt{x}(x\sqrt{x}-1)(\sqrt{x}-1)}\quad\) với \(x>0,\) \(x\) khác \(1\)

ĐKXĐ: x > 0 và x ≠ 1

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

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

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

$= \frac{x + 2 + x - \sqrt{x} + x + \sqrt{x} + 1}{\sqrt{x}(\sqrt{x}-1)(x+\sqrt{x}+1)} \cdot \frac{2}{\sqrt{x}-1}$

$= \frac{3x + 3}{\sqrt{x}(\sqrt{x}-1)(x+\sqrt{x}+1)} \cdot \frac{2}{\sqrt{x}-1}$

$= \frac{6(x+1)}{\sqrt{x}(\sqrt{x}-1)^2(x+\sqrt{x}+1)}$

=> \(N=\left(\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\right)+3\left(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\right)\)

mà ta có bđt: \(\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\ge\frac{\left(a+b+c\right)^2}{2\left(a+b+c\right)}=\frac{a+b+c}{2}=\frac32\)

ta có bđt: \(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\ge\frac{9}{2\left(a+b+c\right)}=\frac32\)

=> \(3\left(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\right)\ge3\cdot\frac32=\frac92\)

=> \(N\ge\frac32+\frac92=6\)

dấu "=" xảy ra khi : a=b=c=1

=> \(N=(a2b+c+b2a+c+c2a+b)+3(1b+c+1a+c+1a+b)N=(b+ca2​+a+cb2​+a+bc2​)+3(b+c1​+a+c1​+a+b1​)\)

mà ta có bđt: \(a2b+c+b2a+c+c2a+b≥(a+b+c)22(a+b+c)=a+b+c2=32b+ca2​+a+cb2​+a+bc2​≥2(a+b+c)(a+b+c)2​=2a+b+c​=23​\)

ta có bđt: \(1b+c+1a+c+1a+b≥92(a+b+c)=32b+c1​+a+c1​+a+b1​≥2(a+b+c)9​=23​\)

=> \(3(1b+c+1a+c+1a+b)≥3⋅32=923(b+c1​+a+c1​+a+b1​)≥3⋅23​=29​\)

=> \(N≥32+92=6N≥23​+29​=6\)

dấu "=" xảy ra khi : a=b=c=1

nhận thấy: \(\left(a^2+1^2\right)\left(1^2+b^2\right)\ge\left(a\cdot1+b\cdot1\right)^2=\left(a+b\right)^2\)

CMTT: \(\Rightarrow\left(b^2+1\right)\left(1+c^2\right)\ge\left(b\cdot1+c\cdot1\right)^2=\left(b+c\right)^2\)

\(\left(c^2+1\right)\left(1^2+a^2\right)\ge\left(c\cdot1+a\cdot1\right)^2=\left(c+a\right)^2\)

=> \(\left\lbrack\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)\right\rbrack^2\ge\left\lbrack\left(a+b\right)\left(b+c\right)\left(c+a\right)\right\rbrack^2\)

vì hai vế đều dương nên căn bậc cả hai vế ta dc

\(\Rightarrow\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)\ge\left(a+b\right)\left(b+c\right)\left(c+a\right)\)

dấu "=" xảy ra khi: \(\frac{a}{1}=\frac{1}{b}\Rightarrow ab=1\)

\(\frac{b}{1}=\frac{1}{c}\Rightarrow bc=1\)

\(\frac{c}{1}=\frac{1}{a}\Rightarrow ac=1\)

=> a=b=c=1

a) \(\Rightarrow\left(\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\right)^2\le\left(1^2+1^2+1^2\right)\left\lbrack\left(\sqrt{a+1}\right)^2+\left(\sqrt{b+1}\right)^2+\left(\sqrt{c+1}\right)^2\right\rbrack\)

\(VT^2\le3\left(a+b+c+3\right)=12\)

=> \(VT\le2\sqrt3<3,5\)

b) => \(VT^2\le3\left(2a+2b+2c\right)=6\)

=> \(VT\le\sqrt6\)

dấu "=" xảy ra khi và chỉ khi:

\(\frac{\sqrt{a+b}}{1}=\frac{\sqrt{b+c}}{1}=\frac{\sqrt{c+a}}{1}\)

=> a=b=c=\(\frac13\)

Đúng nhầm ạ:))

cậu ơi:), có bài gì cậu tung hết một thể cho tớ làm chứ giờ ăn cơm cứ đi hành tớ v

nhận thấy: \(\frac{a+2b+c}{1+a^2}=\frac{\left(a+2b+c\right)\left(1+a^2\right)-a^2\left(a+2b+c\right)}{1+a^2}=a+2b+c-\frac{a^2\left(a+2b+c\right)}{1+a^2}\)

mà \(\frac{a^2\left(a+2b+c\right)}{1+a^2}\le\frac{a^2\left(a+2b+c\right)}{2a}=\frac{a\left(a+2b+c\right)}{2}=\frac{a^2+2ab+ac}{2}\)

=> \(\frac{a+2b+c}{1+a^2}\ge a+2b+c-\frac{a^2+2ab+ac}{2}\)

CMTT:=> \(\frac{b+2c+d}{1+b^2}\ge b+2c+d-\frac{b^2+2bc+bd}{2}\)

\(\frac{c+2d+a}{1+c^2}\ge c+2d+a-\frac{c^2+2dc+ac}{2}\)

\(\frac{d+2a+b}{1+d^2}\ge d+2a+b-\frac{d^2+2ad+bd}{2}\)

=> \(VT\ge4\left(a+b+c+d\right)-\frac12\left(a^2+b^2+c^2+d^2+2ab+2bc+2dc+2ad+ac+bd+ac+bd\right)\) \(VT\ge4\cdot4-\frac12\left(a+b+c+d\right)^2=16-\frac12\cdot16=8\left(đpcm\right)\)

Chắc để tối nay tung lên một thể:))

Final Answer

\(\boxed{\frac{a^{2}}{a + b^{2}} + \frac{b^{2}}{b + c^{2}} + \frac{c^{2}}{c + a^{2}} \geq \frac{a + b + c}{2}}\)

a=b=c=1

ta có: \(\frac{a^3}{a^2+b}=a-\frac{ab}{a^2+b}\ge a-\frac{\sqrt{b}}{2}\)

=> \(VT\ge\left(a+b+c\right)-\frac12\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\)

ta có \(\sqrt{a}=\sqrt{a\cdot1}\le\frac{a+1}{2}\)

=> \(VT\ge\left(a+b+c\right)-\frac{a+b+c+3}{4}=\frac{3\left(a+b+c\right)-3}{4}\)

\(VT\ge\frac{3\cdot3-3}{4}=\frac64=\frac32\)

dấu "=" xảy ra khi a=b=c=1

a=b=c=1