Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
\(A=\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-4\right)\\ A=\left(x^2-5x+4\right)\left(x^2-5x+6\right)\\ A=\left(x^2-5x+5-1\right)\left(x^2-5x+5+1\right)\\ A=\left(x^2-5x+5\right)^2-1\ge-1\)
đẳng thức xảy ra khi :
\(x^2-5x+5=0\\ x^2-2.\dfrac{5}{2}x+\dfrac{25}{4}=\dfrac{25}{4}-5\\ \left(x-\dfrac{5}{2}\right)^2=\dfrac{5}{4}\\ \Rightarrow\left[{}\begin{matrix}x-\dfrac{5}{2}=\sqrt{\dfrac{5}{4}}\\x-\dfrac{5}{2}=-\sqrt{\dfrac{5}{4}}\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=\sqrt{\dfrac{5}{4}}+\dfrac{5}{2}=\dfrac{\sqrt{5}+5}{2}\\x=-\sqrt{\dfrac{5}{4}}+\dfrac{5}{2}=\dfrac{5-\sqrt{5}}{2}\end{matrix}\right.\)
vậy GTNN của A =-1 tại \(\left[{}\begin{matrix}x=\sqrt{\dfrac{5}{4}}+\dfrac{5}{2}=\dfrac{\sqrt{5}+5}{2}\\x=-\sqrt{\dfrac{5}{4}}+\dfrac{5}{2}=\dfrac{5-\sqrt{5}}{2}\end{matrix}\right.\)
\(B=-2x^2-x+5\)
\(=-2\left(x^2+\dfrac{1}{2}x-\dfrac{5}{2}\right)\)
\(=-2\left(x^2+\dfrac{1}{2}x+\dfrac{1}{16}-\dfrac{41}{16}\right)\)
\(=-2\left(x+\dfrac{1}{4}\right)^2+\dfrac{41}{8}\le\dfrac{41}{8}\forall x\)
Dấu " = " xảy ra \(\Leftrightarrow x+\dfrac{1}{4}=0\Leftrightarrow x=-\dfrac{1}{4}\)
Vậy Max B là : \(\dfrac{41}{8}\Leftrightarrow x=-\dfrac{1}{4}\)
\(A=-3x^2+x-2\)
\(=-3\left(x^2-\dfrac{1}{3}x+\dfrac{2}{3}\right)\)
\(=-3\left(x^2-2x.\dfrac{1}{6}+\dfrac{1}{36}+\dfrac{23}{36}\right)\)
\(=-3\left[\left(x-\dfrac{1}{6}\right)^2+\dfrac{23}{36}\right]\)
\(=-3\left(x-\dfrac{1}{6}\right)^2-\dfrac{69}{26}\le-\dfrac{69}{26}\forall x\)
Dấu " = " xảy ra \(\Leftrightarrow x-\dfrac{1}{6}=0\Leftrightarrow x=\dfrac{1}{6}\)
Vậy Max A là : \(\dfrac{-69}{26}\Leftrightarrow x=\dfrac{1}{6}\)
\(B=-2x^2-x+5\)
\(=-2\left(x^2-\dfrac{1}{2}x-\dfrac{5}{2}\right)\)
\(=-2\left(x^2-2x.\dfrac{1}{4}+\dfrac{1}{16}-\dfrac{41}{16}\right)\)
\(=-2\left[\left(x-\dfrac{1}{4}\right)^2-\dfrac{41}{16}\right]\)
\(=-2\left(x-\dfrac{1}{4}\right)^2+\dfrac{41}{8}\le\dfrac{41}{8}\forall x\)
Dấu " = " xảy ra \(\Leftrightarrow x-\dfrac{1}{4}=0\Leftrightarrow x=\dfrac{1}{4}\)
Vậy Max B là : \(\dfrac{41}{8}\Leftrightarrow x=\dfrac{1}{4}\)
\(C=-\left(x+1\right)^2-\left(2x-3\right)^2\)
\(=-x^2-2x-1-4x^2+12x-9\)
\(=-5x^2+10x-10\)
\(=-5\left(x^2-2x+1+1\right)\)
\(=-5\left[\left(x-1\right)^2+1\right]\)
\(=-5\left(x-1\right)^2-5\le-5\forall x\)
Dấu " = " xảy ra \(\Leftrightarrow x-1=0\Leftrightarrow x=1\)
Vậy Max C là : \(-5\Leftrightarrow x=1\)
\(E=2-5x^2-y^2-4xy+2x\)
\(=-\left(4x^2+4xy+y^2\right)-\left(x^2-2x+1\right)+3\)
\(=-\left(2x+y\right)^2-\left(x-1\right)^2+3\le3\forall x\)
Dấu " = " xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}2x+y=0\\x-1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=-2x\\x=1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}y=-2\\x=1\end{matrix}\right.\)
Vậy Max E là : \(3\Leftrightarrow x=1;y=-2\)
\(A=x^2+3x+7\)
\(=x^2+2.1,5x+2,25+4,75\)
\(=\left(x+1,5\right)^2+4,75\ge4,75\)
Vậy \(A_{min}=4,75\Leftrightarrow x=-1,5\)
\(B=2x^2-8x\)
\(=2\left(x^2-4x\right)\)
\(=2\left(x^2-4x+4-4\right)\)
\(=2\left[\left(x-2\right)^2-4\right]\)
\(=2\left(x-2\right)^2-8\ge-8\)
Vậy \(B_{min}=-8\Leftrightarrow x=2\)
( mik k ghi đề nhé bn)
a) (2x)^3 - y^3 + (2x)^3 + y^3 - 16x^3 + 16xy = 16
=> 8x^3 - y^3 + 8x^3 + y^3 - 16x^3 + 16xy = 16
=> 16xy = 16
=> xy = 1
Vì x, y nguyên => x = 1, y = 1 hoặc x = -1, y = -1
mik xin lỗi nha, mik chỉ bt làm câu a
Bài làm:
+ \(C=10\left(x^2-2\right)+5=10x^2-20+5=10x^2-15\ge-15\left(\forall x\right)\)
Dấu "=" xảy ra khi: \(10x^2=0\Rightarrow x=0\)
Vậy \(Min\left(C\right)=-15\Leftrightarrow x=0\)
+ \(D=\left(7-x\right)\left(2x+1\right)=-2x^2+13x+7=-2\left(x^2-\frac{13}{2}x+\frac{169}{16}\right)-\frac{225}{8}\)
\(=-2\left(x-\frac{13}{4}\right)^2-\frac{225}{8}\le-\frac{225}{8}\left(\forall x\right)\)
Dấu "=" xảy ra khi: \(-2\left(x-\frac{13}{4}\right)^2=0\Rightarrow x=\frac{13}{4}\)
Vậy \(Max\left(D\right)=-\frac{225}{8}\Leftrightarrow x=\frac{13}{4}\)
+ \(H=x^2+y^2+2x-4y+10=\left(x^2+2x+1\right)+\left(y^2-4y+4\right)+5\)
\(=\left(x+1\right)^2+\left(y-2\right)^2+5\ge5\left(\forall x,y\right)\)
Dấu "=" xảy ra khi: \(\hept{\begin{cases}\left(x+1\right)^2=0\\\left(y-2\right)^2=0\end{cases}\Rightarrow}\hept{\begin{cases}x=-1\\y=2\end{cases}}\)
Vậy \(Min\left(H\right)=5\Leftrightarrow\hept{\begin{cases}x=-1\\y=2\end{cases}}\)
+ \(E=-x^2-4x+6y-y^2-2021=-\left(x^2+4x+4\right)-\left(y^2-6y+9\right)-2008\)
\(=-\left(x+2\right)^2-\left(y-3\right)^2-2008\le-2008\left(\forall x,y\right)\)
Dấu "=" xảy ra khi: \(\hept{\begin{cases}-\left(x+2\right)^2=0\\-\left(y-3\right)^2=0\end{cases}\Rightarrow}\hept{\begin{cases}x=-2\\y=3\end{cases}}\)
Vậy \(Max\left(E\right)=-2008\Leftrightarrow\hept{\begin{cases}x=-2\\y=3\end{cases}}\)
Học tốt!!!!
Bài 1 :
\(e,x^2+2xy+y^2-2x-2y+1\)
\(=\left(x+y-1\right)^2\)
Bài 2:
\(b,2x^3+3x^2+2x+3=0\)
\(\Leftrightarrow\left(2x^3+2x\right)+\left(3x^2+3\right)=0\)
\(\Leftrightarrow2x\left(x^2+1\right)+3\left(x^2+1\right)=0\)
\(\Leftrightarrow\left(x^2+1\right)\left(2x+3\right)=0\)
\(\Leftrightarrow2x+3=0\left(x^2+1>0\right)\)
\(\Leftrightarrow x=-\dfrac{3}{2}\)
Áp dụng BĐT Cauchy-Schwarz ta có:
\(A=\left(2x+3y\right)^2\le\left(2^2+3^2\right)\left(x^2+y^2\right)\)
\(\Rightarrow\left(2x+3y\right)^2\le13\cdot52\)
\(\Rightarrow\left(2x+3y\right)^2\le676\)
\(\Rightarrow2x+3y\le\sqrt{676}=26\)
Đẳng thức xảy ra khi \(x=-4;y=-6\) hoặc \(x=4;y=6\)
*Lưu ý:\(\left(\left|2x+3y\right|\right)^2=\left|2x+3y\right|^2=\left(2x+3y\right)^2\)