a) Ta có : \(A=\left|x+1\right|+\left|y-2\right|\)

\(\ge\left|x+1+y-2\right|\)

\(=\left|x+y-1\right|=\left|5-1\right|=\left|4\right|=4\)

Dấu "=" xảy ra <=> (x + 1)(y - 2) \(\ge\)0

Vậy Min A = 4 <=>  (x + 1)(y - 2) \(\ge\)0

Đặt \(x-1=t\Rightarrow x=t+1\)

\(A=\dfrac{2\left(t+1\right)^2-6\left(t+1\right)+5}{t^2}=\dfrac{2t^2-2t+1}{t^2}=\dfrac{1}{t^2}-\dfrac{2}{t}+2=\left(\dfrac{1}{t}-1\right)^2+1\ge1\)

\(A_{min}=1\) khi \(t=1\Rightarrow x=2\)

a) Ta có: \(P=\left(\dfrac{x^2-2x}{2x^2+8}-\dfrac{2x^2}{8-4x+2x^2-x^3}\right)\left(1-\dfrac{1}{x}-\dfrac{2}{x^2}\right)\)

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

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

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

\(=\dfrac{x^3-x^2-2x^2+4x+4x^2}{2\left(x-2\right)\left(x^2+4\right)}\cdot\left(1-\dfrac{1}{x}-\dfrac{2}{x^2}\right)\)

\(=\dfrac{x^3+x^2+4x}{2\left(x-2\right)\left(x^2+4\right)}\cdot\dfrac{x^2-x-2}{x^2}\)

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

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

\(A=5+3\left(2x-1\right)^2\)

Vì \(\left(2x-1\right)^2\ge0\) với mọi x

=>\(5+\left(2x-1\right)^2\ge5\)

Vậy GTNN của A là 5 khi x=1/2

ai làm được các bài nữa ko ạ. mình cần gấp lắm

a, \(A=\left|2x-5\right|+\left|2x-12\right|=\left|2x-5\right|+\left|12-2x\right|\ge\left|2x-5+12-2x\right|=7\)

Dấu "=" xảy ra khi \(\left(2x-5\right)\left(12-2x\right)\ge0\Leftrightarrow\frac{5}{2}\le x\le6\)

Vậy Amin=7 khi 5/2 <= x <= 6

b, \(B=\left|3x+6\right|+\left|3x-8\right|=\left|3x+6\right|+\left|8-3x\right|\ge\left|3x+6+8-3x\right|=14\)

Dấu "=" xảy ra khi \(\left(3x+6\right)\left(8-3x\right)\ge0\Leftrightarrow-2\le x\le\frac{8}{3}\)

Vậy...

c, \(C=\left|x-1\right|+\left|x-2\right|+\left|x-3\right|+\left|x-4\right|=\left(\left|x-1\right|+\left|3-x\right|\right)+\left(\left|x-2\right|+\left|4-x\right|\right)\ge\left|x-1+3-x\right|+\left|x-2+4-x\right|=2+2=4\)

Dấu "=" xảy ra khi \(\hept{\begin{cases}\left(x-1\right)\left(3-x\right)\ge0\\\left(x-2\right)\left(4-x\right)\ge0\end{cases}\Leftrightarrow\hept{\begin{cases}1\le x\le3\\2\le x\le4\end{cases}\Leftrightarrow}2\le x\le3}\)

Vậy...

\(A=\left|x+2\right|+\left|x+1\right|+\left|2x-5\right|\ge\left|x+2+x+1\right|+\left|2x-5\right|=\left|2x+3\right|+\left|5-2x\right|\)

\(\ge\left|2x+3+5-2x\right|=\left|8\right|=8\)

Dấu "=" xảy ra \(\Leftrightarrow\)\(\hept{\begin{cases}\left(x+2\right)\left(x+1\right)\ge0\left(1\right)\\\left(2x+3\right)\left(5-2x\right)\ge0\left(2\right)\end{cases}}\)

\(\left(1\right)\)

TH1 : \(\hept{\begin{cases}x+2\ge0\\x+1\ge0\end{cases}\Leftrightarrow\hept{\begin{cases}x\ge-2\\x\ge-1\end{cases}\Leftrightarrow}x\ge-1}\)

TH2 : \(\hept{\begin{cases}x+2\le0\\x+1\le0\end{cases}\Leftrightarrow\hept{\begin{cases}x\le-2\\x\le-1\end{cases}\Leftrightarrow}x\le-2}\)

\(\left(2\right)\)

TH1 : \(\hept{\begin{cases}2x+3\ge0\\5-2x\ge0\end{cases}\Leftrightarrow\hept{\begin{cases}x\ge\frac{-3}{2}\\x\le\frac{5}{2}\end{cases}\Leftrightarrow}\frac{-3}{2}\le x\le\frac{5}{2}}\)

TH2 : \(\hept{\begin{cases}2x+3\le0\\5-2x\le0\end{cases}\Leftrightarrow\hept{\begin{cases}x\le\frac{-3}{2}\\x\ge\frac{5}{2}\end{cases}}}\) ( loại ) 

Vậy GTNN của \(A\) là \(8\) khi \(-1\le x\le\frac{5}{2}\)

... 

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