Bài 2: 

a) Ta có: \(\left|2x-5\right|\ge0\forall x\)

\(\Leftrightarrow-\left|2x-5\right|\le0\forall x\)

\(\Leftrightarrow-\left|2x-5\right|+3\le3\forall x\)

Dấu '=' xảy ra khi \(x=\dfrac{5}{2}\)

\(C=\left(x-5\right)^2+10\)

Ta có: \(\left(x-5\right)^2\ge0\forall x\)

\(\Rightarrow C=\left(x-5\right)^2+10\ge10\forall x\)

Dấu \("="\) xảy ra khi: \(x-5=0\Leftrightarrow x=5\)

Vậy \(Min_C=10\) khi \(x=5\).

∀x là gì vậy bạn chracter debate

\(M=\left|x-1\right|+\left|x-3\right|=\left|x-1\right|+\left|3-x\right|\)

Áp dụng bđt \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\) ta có:

\(M\ge\left|x-1+3-x\right|=\left|2\right|=2\)

Dấu " = " xảy ra khi \(x-1\ge0;3-x\ge0\)

\(\Rightarrow x\ge1;x\le3\)

\(\Rightarrow1\le x\le3\)

Vậy \(MIN_M=2\) khi \(1\le x\le3\)

\(A=|x-2012|+|x-2013|=|x-2012|+|2013-x|\ge|x-2012+2013-x|=1\)

Dấu = xảy ra \(< =>2012\le x\le2013\)

\(|x-2012|+|x-2013|\)    

\(=|x-2012|+|-\left(2013-x\right)|\)   

\(=|x-2012|+|2013-x|\)    

Ta có 

\(|x-2012|+|2013-x|\ge|x-2012+2013-x|\)    

\(|x-2012|+|2013-x|\ge1\)   

Dấu = xảy ra  

\(\Leftrightarrow\left(x-2012\right)\left(2013-x\right)\ge0\)    

TH 1 : 

\(\hept{\begin{cases}x-2012\ge0\\2013-x\le0\end{cases}}\)    

\(\hept{\begin{cases}x\ge2012\\-x\ge-2013\end{cases}}\)    

\(\hept{\begin{cases}x\ge2012\\x\le2013\end{cases}}\)   \(\Rightarrow2012\le x\le2013\)   

TH 2 

\(\hept{\begin{cases}x-2012\le0\\2013-x\le0\end{cases}}\)   

\(\hept{\begin{cases}x\le2012\\-x\le-2013\end{cases}}\)   

\(\hept{\begin{cases}x\le2012\\x\ge2013\end{cases}}\)    \(\Rightarrow x=\varnothing\)    

Vậy min A = 1 khi và chỉ khi \(2012\le x\le2013\)

\(A=\left(x+2\right)^2+\left|x+2\right|+15\)

Ta có:

\(\left(x+2\right)^2\ge0\forall x\)

\(\left|x+2\right|\ge0\forall x\)

\(\Rightarrow\left(x+2\right)^2+\left|x+2\right|\ge0\forall x\)

\(\Rightarrow\left(x+2\right)^2+\left|x+2\right|+15\ge15\forall x\)

\(\Rightarrow A\ge15\)Dấu bằng xảy ra.

\(\Leftrightarrow x+2=0\Leftrightarrow x=-2\)

Vậy \(minA=15\Leftrightarrow x=-2\)

`C=-2x(x+7)=-2x^2-14x`

`=-(2x^2+14x)`

`=-( (\sqrt2x)^2 + 2.\sqrt2 x . (7\sqrt2)/2 + ((7\sqrt2)/2)^2 )+49/2`

`=-(\sqrt2x+(7\sqrt2)/2)^2+49/2`

`=> C_(max) = 49/2 <=> x=-7/2`

`D=-3x^2+5x-9`

`=-(3x^2-5x+9)`

`=-((\sqrt3x)^2 - 2.\sqrt3x . (5\sqrt3)/6 + ((5\sqrt3)/6)^2)-83/12`

`=-(\sqrt3x-(5\sqrt3)/6)^2-83/12`

`=> D_(max)=-83/12 <=> \sqrt3x - (5\sqrt3)/6=0 <=> x=5/6`

a) Ta có: \(C=-2x\left(x+7\right)\)

\(=-2\left(x^2+7x\right)\)

\(=-2\left(x^2+7x+\dfrac{49}{4}-\dfrac{49}{4}\right)\)

\(=-2\left(x+\dfrac{7}{2}\right)^2+\dfrac{49}{2}\le\dfrac{49}{2}\forall x\)

Dấu '=' xảy ra khi \(x=-\dfrac{7}{2}\)

b) Ta có: \(D=-3x^2+5x-9\)

\(=-3\left(x^2-\dfrac{5}{3}x+3\right)\)

\(=-3\left(x^2-2\cdot x\cdot\dfrac{5}{6}+\dfrac{25}{36}+\dfrac{83}{36}\right)\)

\(=-3\cdot\left(x-\dfrac{5}{6}\right)^2-\dfrac{83}{12}\le-\dfrac{83}{12}\forall x\)

Dấu '=' xảy ra khi \(x=\dfrac{5}{6}\)

\(A=\frac{3x^2+8x+6}{x^2+2x+1}\) \(\left(x\ne\pm1\right)\)

\(A=\frac{\left(3x^2+6x+3\right)+\left(2x+3\right)}{\left(x+1\right)^2}\)

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

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

Vì\(\left(x+1\right)^2\ge0\forall x\)

\(\Rightarrow3+\frac{2x+3}{\left(x+1\right)^2}\ge3\Leftrightarrow A\ge3\)

Dấu "="xảy ra khi \(2x+3=0\Rightarrow x=\frac{-3}{2}\)

Gọi k là một giá trị của A ta có: 

\(\frac{\left(3x^2-8x+6\right)}{\left(x^2+2x+1\right)}=k\)

\(\Leftrightarrow3x^2-8x+6=k\left(x^2-2x+1\right)\)

\(\Leftrightarrow\left(3-k\right)x^2-\left(8-2k\right)x+6-k=0\)(*)

Ta cần tìm k để PT (*) có nghiệm 
Xét: \(\Delta=\left(8-2k\right)^2-4\left(3-k\right)\left(6-k\right)=64-32k+4k^2-4\left(18-9k+k^2\right)=4k-8\)

Để PT (*) có nghiệm thì: \(\Delta\ge0\Leftrightarrow4k-8\ge0\Leftrightarrow k\ge2\)

Dấu "=" xảy ra khi: \(-\left(8-2.2\right)x+6-2=0\Leftrightarrow-4x+4=0\Rightarrow x=1\)

Vậy: \(B\ge2\)suy ra: B = 2 khi x = 1