Tìm gia trị nhỏ nhất
A= |x-3|+1
B=|x-2018|+|2019-x|
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a) x2 +x +1 = x2 + x + 1/4 + 3/4 =(x+1/2)2 + 3/4
=> GTNN a) =3/4 khi x=-1/2
b) 4x2 +4x -5 = 4x2 + 4x +1 -6 = (2x+1)2-6
=> GTNN b) = -6 khi x=-1/2
c) (x-3)(x+5) +4 = x2+2x -11 = x2+2x +1-12=(x+1)2-12
GTNN c) =12 khi x=-1
d) x2-4x+y2-8y+6=x2-4x+4+y2-8y+16-14=(x-2)2+(y-4)2-14
GTNN d) =-14 khi x=2 , y=4
\(a,=\left(x^2+x+\dfrac{1}{4}\right)+\dfrac{3}{4}=\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\)
Dấu \("="\Leftrightarrow x=-\dfrac{1}{2}\)
\(b,=\left(4x^2+4x+1\right)-6=\left(2x+1\right)^2-6\ge-6\)
Dấu \("="\Leftrightarrow x=-\dfrac{1}{2}\)
\(c,=x^2+2x-15+4=\left(x+1\right)^2-12\ge-12\)
Dấu \("="\Leftrightarrow x=-1\)
\(d,=\left(x^2-4x+4\right)+\left(y^2-8y+16\right)-14=\left(x-2\right)^2+\left(y-4\right)^2-14\ge-14\)
Dấu \("="\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=4\end{matrix}\right.\)
Ta có: \(A=\left|x-2018\right|+\left|2019-x\right|+\left|x-2020\right|\)
\(A=\left(\left|x-2018\right|+\left|2020-x\right|\right)+\left|2019-x\right|\)
\(\Rightarrow A\ge\left|x-2018+2020-x\right|+\left|2019-x\right|=2+\left|2019-x\right|\)
Dấu "=" xảy ra <=> \(\left(x-2018\right)\left(2020-x\right)\ge0\)
\(\Rightarrow\left(x-2018\right)\left(x-2020\right)\le0\)
\(\Rightarrow\hept{\begin{cases}x-2018\ge0\\x-2020\le0\end{cases}\Rightarrow\hept{\begin{cases}x\ge2018\\x\le2020\end{cases}\Rightarrow}2018\le x\le2020}\)
Và \(\left|2019-x\right|\ge0\), Min (A) = 2 <=> |2019-x| = 0 <=> x= 2019
\(A=\left|2018-x\right|+\left|2019-x\right|+\left|2020-x\right|\)
\(=\left|2018-x\right|+\left|2019-x\right|+\left|x-2020\right|\)
Áp dụng BĐT \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\) :
\(A\ge\left|2018-x+x-2020\right|+\left|2019-x\right|=2+\left|2019-x\right|\ge2\)
Dấu "=" xảy ra \(\Leftrightarrow\left(2018-x\right)\left(x-2020\right)\ge0;2019-x=0\Leftrightarrow x=2019\left(tm\right)\)
Vậy GTNN của A là 2 tại x=2019
\(A=\left(|2018-x|+|2020-x\right)+|2019-x|\)
Đặt \(B=|2018-x|+|2020-x|\)
\(=|2018-x|+|x-2020|\ge|2018-x+x-2020|\)
Hay \(B\ge2\left(1\right)\)
Dấu "=" xảy ra\(\Leftrightarrow\left(2018-x\right)\left(x-2020\right)\ge0\)
\(\Leftrightarrow\hept{\begin{cases}2018-x\ge0\\x-2020\ge0\end{cases}}\)hoặc \(\hept{\begin{cases}2018-x< 0\\x-2020< 0\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}x\le2018\\x\ge2020\end{cases}\left(loai\right)}\)hoặc \(\hept{\begin{cases}x>2018\\x< 2020\end{cases}}\)
\(\Leftrightarrow2018< x< 2020\)
Đặt \(C=|2019-x|\)
Vì \(|2019-x|\ge0;\forall x\)
Hay \(C\ge0;\forall x\left(2\right)\)
Dấu "=" xảy ra\(\Leftrightarrow2019-x=0\)
\(\Leftrightarrow x=2019\)
Từ (1) và (2) \(\Rightarrow B+C\ge2+0\)
Hay \(A\ge2\)
Dấu "=" xảy ra\(\Leftrightarrow\hept{\begin{cases}2018< x< 2020\\x=2019\end{cases}\Leftrightarrow}x=2019\)
Vậy MIN A=2 \(\Leftrightarrow x=2019\)
\(\left|x-\dfrac{2}{3}\right|-4\ge-4\forall x\)
Dấu '=' xảy ra khi \(x=\dfrac{2}{3}\)
\(F=\left|2018-x\right|+\left|2019-x\right|\)
\(=\left|2018-x\right|+\left|x-2019\right|\)
Ta có :
\(\left|2018-x\right|+\left|x-2019\right|\ge\left|2018-x+x-2019\right|\)
=> \(F\ge\left|-1\right|\)
=> \(F\ge1\)
Dấu = xảy ra khi : ( 2018 - x ) ( x - 2019 ) > 0
TH1 : \(\hept{\begin{cases}2018-x>0\\x-2019>0\end{cases}}\)
=> \(\hept{\begin{cases}x< 2018\\x>2019\end{cases}}\)
=> 2019 < x < 2018 ( vô lí - loại )
TH2 : \(\hept{\begin{cases}2018-x< 0\\x-2019< 0\end{cases}}\)
=> \(\hept{\begin{cases}x>2018\\x< 2019\end{cases}}\)
=> 2018 < x < 2019
Vậy giá trị nhỏ nhất của F là 1 khi x thỏa mãn 2018 < x < 2019
Vì \(\left|x-2019\right|\ge0\forall x\)
\(\Rightarrow A\ge2018\forall x\)
Dấu "=" xảy ra \(\Leftrightarrow x-2019=0\Leftrightarrow x=2019\)
Vậy Amin = 2018 <=> x = 2019
Tham khảo nha nhóc
https://olm.vn/hoi-dap/detail/223396249611.html
Tương tự à
Áp dụng BĐT \(\left|x\right|+\left|y\right|\ge\left|x+y\right|\)
\(\Rightarrow A=\left|x+2018\right|+\left|2019-x\right|\ge\left|\left(x+2018\right)+\left(2019-x\right)\right|=4037\)
\(\Rightarrow A_{min}=4037\)(Dấu "="\(\Leftrightarrow x\le2019\))
a)\(A=\left|x-3\right|+1\ge1\left(Với\forall x\in Z\right)\)
Dấu "=" xảy ra khi
x - 3 = 0 <=> x = 3
Vậy Min A = 1 <=> x = 3
b)\(B=\left|x-2018\right|+\left|2019-x\right|\ge\left|x-2018+2019-x\right|=1\)
Dấu "=" xảy ra khi
x = 2018 ; y = 2019
Vậy Min B = 1 <=> x = 2018 ; y = 2019