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x + y = 1 => y = 1 - x mà x,y dương => 0 < x < 1
Suy ra : \(A=2x^2-\left(1-x\right)^2+x+\frac{1}{x}+1=2x^2-1+2x-x^2+x+\frac{1}{x}+1\)
\(=x^2+3x+\frac{1}{x}=x^2-x+\frac{1}{4}+4x+\frac{1}{x}+\frac{1}{4}\)
\(=\left(x-\frac{1}{2}\right)^2+4x+\frac{1}{x}+\frac{1}{4}\)
Mà \(4x+\frac{1}{x}\ge2\sqrt{4x.\frac{1}{x}}=2.2=4\). Dấu "=" xảy ra <=> 4x = 1/x <=> x = 1/2
Với x = 1/2 thì ( x - 1/2 )2 cũng đạt GTNN là 0 => y = 1 - a = 1/2
Vậy min\(A=4+\frac{1}{4}=\frac{17}{4}\)<=> x = y = 1/2
Cách giải như sau
x + y = 1 => y = 1 - x mà x,y dương => 0 < x < 1
Suy ra : A=2x2−(1−x)2+x+1x +1=2x2−1+2x−x2+x+1x +1
=x2+3x+1x =x2−x+14 +4x+1x +14
=(x−12 )2+4x+1x +14
Mà 4x+1x ≥2√4x.1x =2.2=4. Dấu "=" xảy ra <=> 4x = 1/x <=> x = 1/2
Với x = 1/2 thì ( x - 1/2 )2 cũng đạt GTNN là 0 => y = 1 - a = 1/2
Vậy minA=4+14 =174 <=> x = y = 1/2
HOK TỐT
x+y=1=>y=1-x
\(Q=2x^2-y^2+x+\frac{1}{x}+2020\)\(=2x^2-\left(1-x\right)^2+x+\frac{1}{x}+2020\)\(=2x^2-\left(1-2x+x^2\right)+x+\frac{1}{x}+2020\)\(=2x^2-1+2x-x^2+x+\frac{1}{x}+2020\)
\(=\left(x^2+2x+1\right)+\left(x+\frac{1}{x}\right)+2018\)\(=\left(x+1\right)^2+\left(x+\frac{1}{x}\right)+2018\)
Ta có: \(\left(x+1\right)^2\ge0\forall x>0\)
Áp dụng BĐT Cô-si cho 2 số dương \(x\)và \(\frac{1}{x}\):
\(x+\frac{1}{x}\ge2\sqrt{x.\frac{1}{x}}=2\)
\(\Rightarrow Q\ge2+2018=2020\)
Dấu '=' xảy ra \(\Leftrightarrow\hept{\begin{cases}x+1=0\\x=\frac{1}{x}\end{cases}\Leftrightarrow x=-1}\)\(\Rightarrow y=1-\left(-1\right)=2\)
Vậy \(minQ=2020\Leftrightarrow x=-1;y=2\)
\(S=\dfrac{x}{2}+\dfrac{1}{2x}+\dfrac{y}{2}+\dfrac{2}{y}+\dfrac{1}{2}\left(x+y\right)\)
\(S\ge2\sqrt{\dfrac{x}{4x}}+2\sqrt{\dfrac{2y}{2y}}+\dfrac{1}{2}.3=\dfrac{9}{2}\)
Dấu "=" xảy ra khi \(\left(x;y\right)=\left(1;2\right)\)
\(P=\dfrac{x^2+y^2+6}{x+y}=\dfrac{x^2+y^2+2xy+4}{x+y}=\dfrac{\left(x+y\right)^2+4}{x+y}=x+y+\dfrac{4}{x+y}\)
\(P\ge2\sqrt{\left(x+y\right).\dfrac{4}{x+y}}=4\)
\(P_{min}=4\) khi \(x=y=1\)
\(\left(\frac{1}{x}+\frac{1}{y}\right)\sqrt{1+x^2y^2}\)
\(\ge\frac{2}{\sqrt{xy}}\sqrt{1+x^2y^2}=2\sqrt{\frac{1}{xy}+xy}=2\sqrt{\frac{1}{16xy}+xy+\frac{15}{16xy}}\)
\(\ge2\sqrt{2\sqrt{\frac{1}{16xy}\cdot xy}+\frac{15}{4\left(x+y\right)^2}}=2\sqrt{\frac{1}{2}+\frac{15}{4}}=\sqrt{17}\)
Dấu "=" xảy ra tai x=y=1/2
\(1\ge x+\dfrac{1}{y}\ge2\sqrt{\dfrac{x}{y}}\Rightarrow\dfrac{x}{y}\le\dfrac{1}{4}\)
Đặt \(\dfrac{x}{y}=a\Rightarrow0< a\le\dfrac{1}{4}\)
\(P=\dfrac{\left(\dfrac{x}{y}\right)^2-\dfrac{2x}{y}+2}{\dfrac{x}{y}+1}=\dfrac{a^2-2a+2}{a+1}=\dfrac{4a^2-8a+8}{4\left(a+1\right)}=\dfrac{4a^2-13a+3+5\left(a+1\right)}{4\left(a+1\right)}\)
\(P=\dfrac{5}{4}+\dfrac{\left(1-4a\right)\left(3-a\right)}{4\left(a+1\right)}\ge\dfrac{5}{4}\)
Dấu "=" xảy ra khi \(a=\dfrac{1}{4}\) hay \(\left(x;y\right)=\left(\dfrac{1}{2};2\right)\)
x,y>0 => theo bdt AM-GM thì x+y >/ 2 căn (xy)=2 , x^2+y^2 >/ 2xy=2 (do xy=1)
P=(x+y+1)(x^2+y^2)+4/(x+y)
>/ 2(x+y+1)+4/(x+y)=[(x+y)+4/(x+y)]+(x+y+2)
x,y>0=>x+y>0 => theo bdt AM-GM thì P >/ 2.2+2+2=8
minP=8