\(A=\left(-x^2+10xy-25y^2\right)+\left(-y^2-20y-100\right)-50\)

\(=-\left(x^2-10xy+25y^2\right)-\left(y-10\right)^2-50\)

\(=-\left(x-5y\right)^2-\left(y-10\right)^2-50\le-50\)

Dấu '=' xảy ra khi \(\hept{\begin{cases}x-5y=0\\y-10=0\end{cases}\Leftrightarrow\hept{\begin{cases}y=10\\x=5y=50\end{cases}}}\)

Vậy MAX A = -50 khi x=50, y=10

\(B=\left(-x^2+2xy-y^2\right)+\left(-y^2-2.y.\frac{1}{2}-\frac{1}{4}\right)+\frac{5}{4}\)

\(B=-\left(x-y\right)^2-\left(y+\frac{1}{2}\right)^2+\frac{5}{4}\le\frac{5}{4}\)

B lớn nhất bằng \(\frac{5}{4}\) khi \(\hept{\begin{cases}x-y=0\\y+\frac{1}{2}=0\end{cases}}\) hay \(x=y=-\frac{1}{2}\)

Ta có: \(5x^2+10yz\le5\left(x^2+y^2+z^2\right)=9x\left(y+z\right)+18yz\)\(\Leftrightarrow5x^2\le9x\left(y+z\right)+8yz\le9x\left(y+z\right)+2\left(y+z\right)^2\)\(\Leftrightarrow5\left(\frac{x}{y+z}\right)^2-9\left(\frac{x}{y+z}\right)-2\le0\Leftrightarrow\left(\frac{5x}{y+z}+1\right)\left(\frac{x}{y+z}-2\right)\le0\)

\(\Rightarrow\frac{x}{y+z}\le2\)(Do \(\frac{5x}{y+z}+1>0\forall x,y,z>0\)) 

\(\Leftrightarrow x\le2\left(y+z\right)\Leftrightarrow x+y+z\le3\left(y+z\right)\)

\(\Rightarrow P\le\frac{2x}{\left(y+z\right)^2}-\frac{1}{\left(x+y+z\right)^3}\le\frac{4\left(y+z\right)}{\left(y+z\right)^2}-\frac{1}{\left(3y+3z\right)^3}\)

\(=\frac{4}{y+z}-\frac{1}{27\left(y+z\right)^3}\)

Đặt \(\frac{1}{y+z}=t\)thì \(P\le4t-\frac{1}{27}t^3-16+16=-\frac{1}{27}\left(t-6\right)^2\left(t+12\right)+16\le16\)

Vậy MaxP = 16 khi \(\left(x,y,z\right)=\left(\frac{1}{3},\frac{1}{12},\frac{1}{12}\right)\)

3x^3 + 10x^2 - 5 + n chia hết cho đa thức 3x + 1
Đặt 3x^3 + 10x^2 - 5 + n là A
Theo định lý bơ du:
3x+1=0=>x=-1/3
Thay vào A
A=a-4
Để A chia hết 3x+1
thì a-4=0=>a=4

Đặt f(x) = 3x³ + 10x² + a - 5

      g(x) = 3x + 1

      h(x) là thương trong phép chia f(x) cho g(x)

Ta có f(x) chia hết cho g(x) <=> f(x) = g(x).h(x)

<=> 3x³ + 10x² + a - 5 = ( 3x + 1 ).h(x) (1)

Với x = -1/3 

(1) <=> a - 4 = 0 => a = 4

Vậy a = 4 thì f(x) chia hết cho g(x)

\(A=\frac{a^2}{a^2-b^2-c^2}+\frac{b^2}{b^2-a^2-c^2}+\frac{c^2}{c^2-b^2-a^2}\)

\(A=\frac{a^2}{\left(b+c\right)^2-b^2-c^2}+\frac{b^2}{\left(a+c\right)^2-a^2-c^2}+\frac{c^2}{\left(a+b\right)^2-b^2-a^2}\)

\(A=\frac{a^2}{2bc}+\frac{b^2}{2ac}+\frac{c^2}{2ab}\)

\(A=\frac{a^3+b^3+c^3}{2abc}\)

Có \(a^3+b^3+c^3=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)+3abc=3abc\)nên

\(A=\frac{3}{2}\).

Giúp mình nha!!!!!

Như nài cx được nè :3 

\(A=\left(x^2-3x+1\right)\left(x^2+2x+1\right)-6x^2\)

\(=\left(x^4+2x^3+x^2-3x^3-6x^2-3x+x^2+2x+1\right)-6x^2\)

\(=\left(x^4-x^3-4x^2-x+1\right)-\left(\sqrt{6}x\right)^2\)

\(=\left(x^4-x^3-4x^2-x+1-\sqrt{6}x\right)\left(x^4-x^3-4x^2-x+1+\sqrt{6}x\right)\)

Ta có : 2a² + 2b² = 5ab

=> 2a² + 2b² - 4ab = 5ab - 4ab

=> 2(a² + b² - 2ab) = ab

=> (a - b)² = ab/2 

Lại có 2a² + 2b² = 5ab

=> 2a² + 2b² + 4ab = 5ab + 4ab

=> 2(a + b)² = 9ab

=> (a + b)² = 9ab/2

Ta có P² = \(\left(\frac{a+b}{a-b}\right)^2=\frac{\left(a+b\right)^2}{\left(a-b\right)^2}=\frac{\frac{9ab}{2}}{\frac{ab}{2}}=9\)

=> P = \(\pm\)3

Vậy P = \(\pm\)3

+)\(x^2-2y^2=xy\)

\(2y^2=x^2-xy\)

\(2y^2=x.\left(x-y\right)\)

\(\Rightarrow x-y=\frac{2y^2}{x}\left(1\right)\)

+)\(x^2-2y^2=xy\)

\(x^2=xy+2y^2\)

\(x^2=xy+2y^2-y^2+y^2\)

\(x^2=xy+y^2+y^2\)

\(x^2=\left(x+y\right).y+y^2\)

\(\Rightarrow x^2-y^2=\left(x+y\right).y\)

\(\Rightarrow x+y=\frac{x^2-y^2}{y}\left(2\right)\)

+)Từ (1) và (2)

\(\Rightarrow A=\frac{x-y}{x+y}=\frac{\frac{2y^2}{x}}{\frac{x^2-y^2}{y}}\)

\(\Rightarrow A=\frac{2y^2}{x}:\frac{x^2-y^2}{y}\)

\(\Rightarrow A=\frac{2y^3}{x^3-x.y^2}\)

Chúc bạn học tốt

\(x-y=5\Leftrightarrow x=5+y\).

\(\frac{x-3y}{5-2y}=\frac{5+y-3y}{5-2y}=\frac{5-2y}{5-2y}=1\)