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1) \(a^2+\frac{1}{a^2}=14\Leftrightarrow a^2+\frac{1}{a^2}+2a.\frac{1}{a}=16\Leftrightarrow\left(a+\frac{1}{a}\right)^2=16\Rightarrow a+\frac{1}{a}=4\)
\(\Rightarrow\left(a+\frac{1}{a}\right)\left(a^2+\frac{1}{a^2}\right)=a^3+\frac{1}{a}+a+\frac{1}{a^3}=a^3+4+\frac{1}{a^3}=4.14=56\)
\(\Rightarrow a^3+\frac{1}{a^3}=52\)
Ta có : \(\left(a^2+\frac{1}{a^2}\right)\left(a^3+\frac{1}{a^3}\right)=a^5+\frac{1}{a}+a+\frac{1}{a^5}=a^5+4+\frac{1}{a^5}=14.52\)
\(\Rightarrow a^5+\frac{1}{a^5}=14.52-4=724\)
2) \(A=2xy-x^2-4y^2+2x+10y-2000\)
\(=\left(-x^2+2xy-y^2\right)+\left(2x-2y\right)+\left(-3y^2+12y-12\right)-1988\)
\(=-\left(x-y\right)^2+2\left(x-y\right)-1-3\left(y^2-4y+4\right)-1987\)
\(=-\left(x-y-1\right)^2-3\left(y-2\right)^2-1987\le-1987\forall x;y\) có GTLN là 2013
Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}x-y-1=0\\y-2=0\end{cases}\Rightarrow\hept{\begin{cases}x=3\\y=2\end{cases}}}\)
Vậy \(A_{max}=-1987\) tại \(x=3;y=2\)
\(A=x^2+2x\left(y+1\right)+\left(y+1\right)^2-\left(y+1\right)^2+2y^2-4y+2028\)
\(=\left(x+y+1\right)^2-y^2-2x-1+2y^2-4y+2028\)
\(=\left(x+y+1\right)^2-6x+y^2+2027\)
\(=\left(x+y+1\right)+\left(y-3\right)^2+2018\ge2018\forall x;y\) (do...)
=> MinA = 2018 \(\Leftrightarrow\left\{{}\begin{matrix}x+y=-1\\y=3\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=-4\\y=3\end{matrix}\right.\)
\(A=x^2+2xy+2y^2+2x-4y+2013\)
\(=\left(x^2+y^2+1+2x+2y+2xy\right)-1-2y+y^2-4y+2013\)\(=\left(x+y+1\right)^2+\left(y^2-2.y.3+9\right)-9+2012\)
\(=\left(x+y+1\right)^2+\left(y-3\right)^2+2003\)
mà \(\left(x+y+1\right)^2,\left(y-3\right)^2\ge0\)
\(\Rightarrow A=x^2+2xy+2y^2+2x-4y+2013=\left(x+y+1\right)^2+\left(y-3\right)^2+2003\ge2003\)
\(\Rightarrow Min\left(A\right)=2003\)
a) \(A=4x^2-12x+100=\left(2x\right)^2-12x+3^2+91=\left(2x-3\right)^2+91\)
Ta có: \(\left(2x-3\right)^2\ge0\forall x\inℤ\)
\(\Rightarrow\left(2x-3\right)^2+91\ge91\)
hay A \(\ge91\)
Dấu "=" xảy ra <=> \(\left(2x-3\right)^2=0\)
<=> 2x-3=0
<=> 2x=3
<=> \(x=\frac{3}{2}\)
Vậy Min A=91 đạt được khi \(x=\frac{3}{2}\)
b) \(B=-x^2-x+1=-\left(x^2+x-1\right)=-\left(x^2+x+\frac{1}{4}-\frac{5}{4}\right)=-\left(x+\frac{1}{2}\right)^2+\frac{5}{4}\)
Ta có: \(-\left(x+\frac{1}{2}\right)^2\le0\forall x\)
\(\Rightarrow-\left(x+\frac{1}{2}\right)^2+\frac{5}{4}\le\frac{5}{4}\) hay B\(\le\frac{5}{4}\)
Dấu "=" \(\Leftrightarrow-\left(x+\frac{1}{2}\right)^2=0\)
\(\Leftrightarrow x+\frac{1}{2}=0\)
\(\Leftrightarrow x=\frac{-1}{2}\)
Vậy Max B=\(\frac{5}{4}\)đạt được khi \(x=\frac{-1}{2}\)
\(C=2x^2+2xy+y^2-2x+2y+2\)
\(C=x^2+2x\left(y-1\right)+\left(y-1\right)^2+x^2+1\)
\(\Leftrightarrow C=\left(x+y-1\right)^2+x^2+1\)
Ta có:
\(\hept{\begin{cases}\left(x+y-1\right)^2\ge0\forall x;y\inℤ\\x^2\ge0\forall x\inℤ\end{cases}}\)
\(\Leftrightarrow\left(x+y-1\right)^2+x^2+1\ge1\)
hay C\(\ge\)1
Dấu "=" xảy ra khi \(\hept{\begin{cases}\left(x+y-1\right)^2=0\\x^2=0\end{cases}\Leftrightarrow\hept{\begin{cases}x+y=1\\x=0\end{cases}\Leftrightarrow}\hept{\begin{cases}y=1\\x=0\end{cases}}}\)
Vậy Min C=1 đạt được khi y=1 và x=0
\(A=\left(x^2+y^2+1+2xy+2x+2y\right)+\left(y^2-6y+9\right)+2018\)
\(A=\left(x+y+1\right)^2+\left(y-3\right)^2+2018\ge2018\)
\(A_{min}=2018\) khi \(\left\{{}\begin{matrix}x=-4\\y=3\end{matrix}\right.\)
Giúp mk bài hình mk mới đăng với Nguyễn Việt Lâm Quản lý, ý b,c, d thôi
A = [y^2 +2y(x-2) + (x-2)^2] + (x^2-6x+9) + 1
= (y+x-2)^2 + (x-3)^2 + 1 >=1
Dấu = xảy ra khi <=> y+x-2 = x-3=0
<=> x=3; y=-1