\(A=\frac{\left|x-2017\right|+2018}{\left|x-2017\right|+2019}\)

\(A=\frac{\left|x-2017\right|+2019-1}{\left|x-2017\right|+2019}\)

\(A=1-\frac{1}{\left|x-2017\right|+2019}\)

A nhỏ nhất khi \(1-\frac{1}{\left|x-2017\right|+2019}\)nhỏ nhất

khi \(\frac{1}{\left|x-2017\right|+2019}\)lớn nhất

khi \(\left|x-2017\right|+2019\)nhỏ nhất

mà |x - 2017| \(\ge0\)

=> |x - 2017| + 2019 \(\ge2019\)

Vậy A nhỏ nhất khi A = 2019 khi x - 2017 = 0 => x = 2017

\(A=\frac{\backslash x-2017\backslash+2018}{\backslash x-2017\backslash+2019}\) 

\(A=\frac{2018}{2019}\)

Ta có : \(A=\frac{\left|x-2017\right|+2018}{\left|x-2017\right|+2019}=\frac{\left|x-2017\right|+2019-1}{\left|x-2017\right|+2019}\)

\(=1-\frac{1}{\left|x-2017\right|+2019}\)

Ta có : \(\left|x-2017\right|\ge0\)

\(\Rightarrow\left|x-2017\right|+2019\ge2019\)

\(\Rightarrow\frac{1}{\left|x-2017\right|+2019}\le\frac{1}{2019}\)

\(\Rightarrow-\frac{1}{\left|x-2017\right|+2019}\ge-\frac{1}{2019}\)

\(\Rightarrow1-\frac{1}{\left|x-2017\right|+2019}\ge1-\frac{1}{2019}=\frac{2018}{2019}\)

Hay : \(A\ge\frac{2018}{2019}\)

Dấu "=" xảy ra \(\Leftrightarrow\left|x-2017\right|=0\Leftrightarrow x=2017\)

Vậy : min \(A=\frac{2018}{2019}\) tại \(x=2017\)

Tìm giá trị nhỏ nhất của:P=/x-2016/+/x-2017/.

Áp dụng BĐT /a+b/. ≤/a/+/b/. ⇒ P=/x-2016/+/x-2017/= /x-2016/+/2017-x/ lớn hơn hoặc bằng /x-2016+2017-x/=1.

Vậy GTNN của P là 1 <=> 0. ≤(x-2016)(2017-x) <=> 2016. ≤x. ≤2017. 

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\(C=\frac{\left|x-2017\right|+2018}{\left|x-2017\right|+2019}=\frac{\left|x-2017\right|+2019-1}{\left|x-2017\right|+2019}=1-\frac{1}{\left|x-2017\right|+2019}\)

C nhỏ nhất => \(\frac{1}{\left|x-2017\right|+2019}\)lớn nhất 

=> |x+2017|+2019 nhỏ nhất

\(\left|x+2017\right|\ge0\Rightarrow\left|x+2017\right|+2019\ge2019\)

dấu = xảy ra khi |x+2017|=0 

=> x=-2017

Vậy MIN C=\(\frac{2018}{2019}\)

p/s: :)) có vẻ ko hoàn hảo lắm 

GTNN là 2019 nhé 

\(Q=\left|x-2017\right|+\left|x-2018\right|+\left|x-2019\right|\)

       \(\ge\left|x-2018\right|+\left|x-2017+2019-x\right|\)

        \(\ge\left|x-2018\right|+2\ge2\)

Dấu "=" <=> x = 2018

\(Q=\left|x-2017\right|+\left|x-2018\right|+\left|2019-x\right|\)

\(\ge x-2017+0+2019-x=2\)

Dấu "=" xảy ra khi \(\hept{\begin{cases}2017\le x\le2019\\x=2018\end{cases}}\Leftrightarrow x=2108\) (thỏa mãn cả hai trường hợp)

Vậy...

P/s: Ở đây mình gộp hai trường hợp \(x-2017\ge0;2019-x\ge0\) thành \(2017\le x\le2019\) cho lẹ nha!

Ta có: \(\frac{x-2019}{2018}+\frac{x-2018}{2017}=\frac{x-2017}{2016}+\frac{x-2016}{2015}\)

\(\Leftrightarrow\left(\frac{x-2019}{2018}+1\right)+\left(\frac{x-2018}{2017}+1\right)=\left(\frac{x-2017}{2016}+1\right)+\left(\frac{x-2016}{2015}+1\right)\)

\(\Leftrightarrow\frac{x-1}{2018}+\frac{x-1}{2017}=\frac{x-1}{2016}+\frac{x-1}{2015}\)

\(\Leftrightarrow\frac{x-1}{2018}+\frac{x-1}{2017}-\frac{x-1}{2016}-\frac{x-1}{2015}=0\)

\(\Leftrightarrow\left(x-1\right)\left(\frac{1}{2018}+\frac{1}{2017}-\frac{1}{2016}-\frac{1}{2015}\right)=0\)

\(\Leftrightarrow x-1=0\)( vì \(\frac{1}{2018}+\frac{1}{2017}-\frac{1}{2016}-\frac{1}{2015}\ne0\))

\(\Leftrightarrow x=1\)

Vạy x=1

M = | x - 2019 | + | x - 2018 | - 2017

M = | x - 2019 | + | x - 2018 | - 2017 \(\ge\)- 2017

Dấu " = " xảy ra \(\Leftrightarrow\)x - 2019 = 0 hoặc x - 2018 = 0

\(\Rightarrow\)x = 2019 hoặc x = 2018

Min M = - 2017 \(\Leftrightarrow\)x = 2019 hoặc x = 2018

*) Ta chứng minh bổ đề: \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\)

\(\Leftrightarrow\left(\left|a\right|+\left|b\right|\right)^2\ge\left(\left|a+b\right|\right)^2\)

\(\Leftrightarrow a^2+b^2+2\left|ab\right|\ge a^2+b^2+2ab\)

\(\Leftrightarrow2\left|ab\right|\ge2ab\) 

\(\Leftrightarrow\left|ab\right|\ge ab\) ( luôn đúng ) 

Dấu "=" xảy ra khi \(ab\ge0\)

Theo bài cho: M = |x-2019| + |x-2018| - 2017

=> M = |x - 2019| + |2018 - x| - 2017

Áp dụng bổ đề trên => | x - 2019 | + | 2018 - x| \(\ge\) | x - 2019 + 2018 - x |

=> | x - 2019 | + | 2018 - x | \(\ge\)1

=> | x - 2019 | + | 2018 - x | - 2017 \(\ge\)1 - 2017

=> M \(\ge\)-2016

Dấu "=" xảy ra khi ( x - 2019 ).( 2018 - x)\(\ge\)0

Ta xét 2 trường hợp:

+) Nếu \(\hept{\begin{cases}x-2019\ge0\\2018-x\ge0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\ge2019\\x\le2018\end{cases}}\)( loại )

+) Nếu \(\hept{\begin{cases}x-2019\le0\\2018-x\le0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\le2019\\x\ge2018\end{cases}}\)\(\Leftrightarrow2018\le x\le2019\)( thỏa mãn )

Vạy: GTNN của M = -2016 khi \(2018\le x\le2019\)

Đặt: \(\left|x-2017\right|=t\ge0\) ta có: \(l=\frac{t+2017}{t+2018}=\frac{t+2018-1}{t+2018}=1-\frac{1}{t+2018}\ge1-\frac{1}{2018}=\frac{2017}{2018}\)

Dấu "=" xảy ra khi: \(t=0\Leftrightarrow x=2017\)

Đặt: |x−2017|=t≥0 ta có: l=t+2017t+2018 =t+2018−1t+2018 =1−1t+2018 ≥1−12018 =20172018 

Dấu "=" xảy ra khi: t=0⇔x=2017

 ...

..

\(A=\frac{\left|x-2017\right|+2017}{\left|x-2017\right|+2018}=1-\frac{1}{\left|x-2017\right|+2018}\)

A bé nhất khi \(\frac{1}{\left|x-2017\right|+2018}\) lớn nhất.

Mà \(\frac{1}{\left|x-2018\right|+2018}\le\frac{1}{2018}\forall x\) (do \(\left|x-2018\right|\ge0\forall x\))

Suy ra \(A\ge1-\frac{1}{2018}=\frac{2017}{2018}\)

Dấu "=" xảy ra \(\Leftrightarrow\left|x-2017\right|=0\Leftrightarrow x=2017\)

Vậy \(A_{min}=\frac{2017}{2018}\Leftrightarrow x=2017\)