Question

Timg GTNN của các bt:

a. \(A=\left|x-2000\right|+\left|x-2001\right|+\left|x-2002\right|\)

b. \(\sqrt{\left(x-8\right)^2}+\sqrt{\left(x-9\right)^2}+\sqrt{\left(x-10\right)^2}+\sqrt{\left(x+11\right)^2}\)

Answer 1

a) \(\left|x-2000\right|+\left|x-2002\right|=\left|x-2000\right|+\left|2002-x\right|\)

\(\ge\left|x-2000+2002-x\right|=2\) (1)

Dấu "=" \(\Leftrightarrow\left(x-2000\right)\left(2002-x\right)\ge0\)

\(\Leftrightarrow2000\le x\le2002\)

+ \(\left|x-2001\right|\ge0\forall x\). "=" \(\Leftrightarrow x=2001\) (2)

Từ (1) và (2) suy ra \(A\ge2\)

Dấu "=" \(\Leftrightarrow x=2001\)

b) \(B=\left|x-8\right|+\left|x-9\right|+\left|x-10\right|+\left|x+11\right|\)

+ \(\left|x-10\right|+\left|x+11\right|=\left|x+11\right|+\left|10-x\right|\)

\(\ge\left|x+11+10-x\right|=21\) (3)

Dấu "=" \(\Leftrightarrow\left(x+11\right)\left(10-x\right)\ge0\Leftrightarrow-11\le x\le10\)

+ \(\left|x-8\right|+\left|x-9\right|\ge\left|x-8+9-x\right|=1\) (4)

"=" \(\Leftrightarrow\left(x-8\right)\left(9-x\right)\ge0\Leftrightarrow8\le x\le9\)

Từ (3) và (4) suy ra \(B\ge22\)

"=" \(\Leftrightarrow8\le x\le9\)