Question

1, Tìm GTNN:

a) A = |x - 1| + |x - 2| + |4 - 2x|

b) B = |x - 1| + |x - 2| + |x - y + 1| + |x - 5|

2, Tìm GTLN:

a) A = \(\dfrac{5}{2x^2+4x+3}\)

b) B = \(\dfrac{8}{2x^2.y^2+2xy+1}\)

Answer 1

\(A=\left|x-1\right|+\left|x-2\right|+\left|4-2x\right|\)

\(\)Áp dụng BĐT: \(\left|A\right|+\left|B\right|\ge\left|A+B\right|\)

\(\Rightarrow A\ge\left|x-1+x-2+4-2x\right|\)

\(\Rightarrow A\ge\left|2x-2x-1-2+4\right|\)

\(\Rightarrow A\ge1\)

Dấu "=" xảy ra khi:

\(\left[{}\begin{matrix}\left\{{}\begin{matrix}x-1< 0\Rightarrow x< 1\\x-2< 0\Rightarrow x< 2\\4-2x< 0\Rightarrow4< 2x\Rightarrow2< x\end{matrix}\right.\\\left\{{}\begin{matrix}x-1\ge0\Rightarrow x\ge1\\x-2\ge0\Rightarrow x\ge2\\4-2x\ge0\Rightarrow4\ge2x\Rightarrow2\ge x\end{matrix}\right.\end{matrix}\right.\)

\(\Rightarrow x=2\)