Question

tính GTNN của: \(\dfrac{\left|x+2017\right|+2018}{\left|x+2017\right|+2019}\)

Answer 1

vì |x+2017|\(\ge\)0

=> |x+2017|+2018\(\ge\)2018

|x+2017|+2019\(\ge\)2019

=> GTNN của \(\dfrac{\left|x+2017\right|+2018}{\left|x+2017\right|+2019}\)=\(\dfrac{2018}{2019}\)

Answer 2

Đặt \(t=\left|x+2017\right|\ge0\)

Đặt biểu thức là T, ta có:

\(T=\dfrac{t+2018}{t+2019}=\dfrac{t+2019-1}{t+2019}=1-\dfrac{1}{t+2019}\)

Ta có: \(t\ge0\Rightarrow t+2019\ge2019\)

\(\Rightarrow\dfrac{1}{t+2019}\le\dfrac{1}{2019}\)

\(\Rightarrow-\dfrac{1}{t+2019}\ge-\dfrac{1}{2019}\)

\(\Rightarrow T\ge1-\dfrac{1}{2019}=\dfrac{2008}{2009}\)

GTNN của T là \(\dfrac{2008}{2009}\) khi \(t=0\Leftrightarrow\left|x+2017\right|=0\Leftrightarrow x=-2017\)