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ap dung bdt \(|a|+|b|\ge|a+b|\) voi \(a.b\ge0\)
thi \(A\ge|x-2016+2007-x|=|1|=1\)
vay GTNN cua A = 1 . Dat duoc khi \(\left(x-2016\right)\left(2017-x\right)\ge0\)
<=> \(2016\le x\le2017\)
chuc ban hoc tot
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1/
l2x+3l=x+2(1)
ta co l2x+3l=\(\hept{\begin{cases}2x+3voix\ge\frac{-3}{2}\\-2x-3voix< \frac{-3}{2}\end{cases}}\)
TH1: neu x>= -3/2 thi (1) <=>2x+3=x+2=>x=-1(chon)
TH2: neu x<= -3/2 thi (1) <=> -2x-3=x+2=>-3x=5=>x=-5/3(chon)
2/
de A dat gtnn thi lx-2006l va l2007l dat gtnn
ma lx-2006l va l2007-xl >=0
=> gtnn cua lx-2006l=0;l2007-xl=0
=> x=2006 hoac 2007
=> gtnn A=1
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'THAM KHẢO
a,
Điều kiện: x+2≥0⇔x≥−2x+2≥0⇔x≥-2
|2x+3|=x+2|2x+3|=x+2
⇔[2x+3=x+22x+3=−x−2⇔[2x+3=x+22x+3=−x−2
⇔[x=−13x=−5⇔[x=−13x=−5
⇔⎡⎣x=−1(t/m)x=−53(t/m)⇔[x=−1(t/m)x=−53(t/m)
Vậy x∈{−1;−53}x∈{-1;-53}
b,
A=|x−2006|+|2007−x|≥|x−2006+2007−x|=|1|=1A=|x−2006|+|2007−x|≥|x−2006+2007−x|=|1|=1
Đẳng thức xảy ra ⇔(x−2006)(2007−x)≥0⇔(x−2006)(2007−x)≥0
⇔(x−2006)(x−2007)≤0⇔(x−2006)(x−2007)≤0
Vì x−2006>x−2007x−2006>x−2007
⇒{x−2006≥0x−2007≤0⇒{x−2006≥0x−2007≤0
⇔{x≥2006x≤2007⇔{x≥2006x≤2007
⇔2006≤x≤2007⇔2006≤x≤2007
Vậy Amin=1⇔2006≤x≤2007
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Áp dụng BĐT |a|+|b|>=|a+b| ta có:
\(\left|x-2006\right|+\left|2007-x\right|\ge\left|x-2006+2007-x\right|=1\)
\(\Rightarrow A\ge1\)
Dấu "=" xảy ra khi \(\hept{\begin{cases}\left|x-2006\right|=0\\\left|2007-x\right|=0\end{cases}\Leftrightarrow}\hept{\begin{cases}x=2006\\x=2007\end{cases}}\)
Vậy MinA=1<=>x=2006 hoặc x=2007
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\(A=\left|x-2006\right|+\left|2007-x\right|\ge\left|x-2006+2007-x\right|=\left|1\right|=1\)
\(minA=1\Leftrightarrow\left(x-2006\right)\left(2007-x\right)\ge0\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x-2006\ge0\\2007-x\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}x-2006\le0\\2007-x\le0\end{matrix}\right.\end{matrix}\right.\) \(\Leftrightarrow2006\le x\le2007\)
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\(A=\left|x-2006\right|+\left|2007-x\right|\)
Vì \(x>2007\) nên \(2x-4013>4014-4013=1\)
\(\Rightarrow A>1\)
Vậy \(A_{min}=1\Leftrightarrow2006\le x\le2007\)
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Ta có : A = |x - 2006| + |2007 - x| ≥ |x - 2006 + 2007 - x|
= |(x - x) - 2006 + 2007| = |1| = 1
Dấu "=" xảy ra khi (x - 2006)(2007 - x) ≥ 0 => 2006 ≤ x ≤ 2007
Vậy gtnn của A là 1 tại 2006 ≤ x ≤ 2007