Mysterious Person

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1.

Đặt \(x+y=a\Rightarrow y=a-x\)

\(\Rightarrow x^2+2x\left(a-x\right)-14\left(a-x\right)-10x+3\left(a-x\right)^2+27=0\)

\(\Leftrightarrow2x^2-4\left(a+1\right)x+3a^2-10a+27=0\)

\(\Delta'=4\left(a+1\right)^2-2\left(3a^2-10a+27\right)\ge0\)

\(\Leftrightarrow-a^2+14a-25\ge0\)

\(\Rightarrow7-2\sqrt{6}\le a\le7+2\sqrt{6}\)

\(\Rightarrow-10-2\sqrt{6}\le P\le-10+2\sqrt{6}\)

2. Chắc đề là \(a;b>0\) (đảm bảo mẫu dương) chứ ko phải \(a.b>4\)

\(M\ge\dfrac{\left(a+b\right)^2}{a+b-8}=\dfrac{\left(a+b-8+8\right)^2}{a+b-8}=\dfrac{\left(a+b-8\right)^2+16\left(a+b-8\right)+64}{a+b-8}\)

\(M\ge a+b-8+\dfrac{64}{a+b-8}+16\ge2\sqrt{\dfrac{64\left(a+b-8\right)}{a+b-8}}+16=32\)

Dấu "=" xảy ra khi \(a=b=8\)

A = -x² - 2xy - y² - 2y² + 10x + 10y + 4y - 25 + 7
= (-x² - 2xy - y² + 10x + 10y - 25) - 2y² + 4y + 7
= -(x² + 2xy + y² - 10x - 10y + 25) - (2y² - 4y - 7)
= -[(x+y)² - 10(x+y) + 25] - (2y² - 4y + 2 - 9)
= -(x + y - 5)² - 2(y² - 2y + 1) + 9
= -(x + y - 5)² - 2(y - 1)² + 9 ≤ 9
Dấu ''='' xảy ra <=> x + y - 5 = 0 và y -1 =0
<=> x + y = 5 và y = 1
<=> x = 4 và y = 1
Vậy max A = 9 <=> x = 4 và y = 1 .
- Mình chúc bạn học tốt nhé !

-2A=2x²+6y²+4xy-20x-28y+36

=(x²+4xy+4y²)+(x²-20x+100)+2(y²-14y+49)-162

=(x+2y)²+(x-10)²+2(y-7)²-162\(\ge\)-162

=> A\(\le81\)

Dấu "=" xảy ra khi

A = -x² - 3y² - 2xy + 10x + 14y - 18

A = -x² - y² -25 + 10x +10y -2xy -2y² + 4y -2 + 9

A = -(x² + y² + ( -5 )² - 10x - 10y + 2xy ) - 2 (y² - 2y + 1 )  + 9

A = -( x + y - 5 )² - 2 ( y - 1 )² + 9 

-( x + y - 5 )²  \(\le\)0 ; - 2 ( y - 1 )² \(\le\)0

\(\Rightarrow\)A  \(\le\)0 + 0 + 9 = 9

Dấu " = " xảy ra \(\Leftrightarrow\)\(\hept{\begin{cases}x+y-5=0\\y-1=0\end{cases}\Rightarrow\hept{\begin{cases}x=4\\y=1\end{cases}}}\)

\(A=\left(-x^2-2xy-y^2\right)-2y^2+\left(10x+10y\right)+4y-18\)

\(=-\left(x+y\right)^2+2\left(x+y\right).5-\left(2y^2-4y+2\right)-16\)

\(=-\left[\left(x+y\right)^2-2\left(x+y\right).5+5^2\right]-2\left(y-1\right)^2+9\)

\(=-\left(x+y-5\right)^2-2\left(y-1\right)^2+9\le9\forall x;y\)

Dấu "=" xảy ra khi \(\hept{\begin{cases}x+y-5=0\\y=1\end{cases}}\Leftrightarrow\hept{\begin{cases}x=5-y\\y=1\end{cases}}\Leftrightarrow\hept{\begin{cases}x=4\\y=1\end{cases}}\)

Vậy \(A_{max}=9\Leftrightarrow\hept{\begin{cases}x=4\\y=1\end{cases}}\)

ko biết

bít chết liền

\(x^2-3y^2-2xy+10x+14y-18\)

\(=x^2-2xy+y^2-2x^2+10x-4y^2+14y-18\)

\(=x^2-2xy+y^2-2\left(x^2-5x+25\right)-4\left(y^2-\frac{7}{2}y+\frac{49}{16}\right)+\frac{177}{4}\)

\(=\left(x-y\right)^2-2\left(x-5\right)^2-4\left(y-\frac{7}{4}\right)^2+\frac{177}{4}\)

.....

A chỉ đạt max

B=(x^2+y^2+1-2xy+2x-2y)+(x^2-4x+4)-10

B=(x-y+1)^2+(x-2)^2-10\(\ge\)-10

C=((x^2+y^2-2xy)-10(x-y)+25)+3(y^2-2y+1)+4

C=(x-y-5)^2+3(y-1)^2+4\(\ge\)4