A+1=(27-12x)/(x^2+9)+1

A+1=(x^2-12x+36)/(x^2+9)

A+1=(x-6)^2/(x^2+9)>=0

Min A+1=0

Min A=-1

Dấu = xảy ra khi và chỉ khi x=6

4-A=4-(27-12x)/(x^2+9)

4-A=(4x^2+36-27+12x)/(x^2+9)

4-A=(4x^2+12x+9)/(x^2+9)

4-A=(2x+3)^2/(x^2+9)

A=4-(2x+3)^2/(x^2+9)<=4

Max A=4 

Dấu = xảy ra khi và chỉ khi x=-3/2 

a)Ta có:

 \(a+b+ab=a^2+b^2\).

\(\Leftrightarrow a^2-ab+b^2=a+b\).

Ta có:

\(P=a^3+b^3+2020\).

\(P=\left(a+b\right)\left(a^2-ab+b^2\right)+2020\).

\(P=\left(a+b\right)\left(a+b\right)+2020\)(vì \(a^2-ab+b^2=a+b\)).

\(P=\left(a+b\right)^2+2020\).

Ta có:

\(\left(a+b\right)^2\ge0\forall a;b\).

\(\Rightarrow\left(a+b\right)^2+2020\ge2020\forall a;b\).

\(\Rightarrow P\ge2020\).

Dấu bằng xảy ra.

\(\Leftrightarrow\hept{\begin{cases}a+b+ab=a^2+b^2\\\left(a+b\right)^2=0\end{cases}}\Leftrightarrow a=b=0\).

Vậy \(maxP=2020\Leftrightarrow a=b=0\).

b)\(A=\frac{27-12x}{x^2+9}\).

Vì \(x^2+9>0\forall x\)nên \(A\)luôn được xác định.

 \(A=\frac{27-12x}{x^2+9}=\frac{4x^2-4x^2+27-12x}{x^2+9}=\frac{\left(4x^2+36\right)-\left(4x^2+12x+9\right)}{x^2+9}\)

\(A=\frac{4\left(x^2+9\right)-\left(2x+3\right)^2}{x^2+9}=4-\frac{\left(2x+3\right)^2}{x^2+9}\).

Ta có:

\(\left(2x+3\right)^2\ge0\forall x\).

\(\Rightarrow\frac{\left(2x+3\right)^2}{x^2+9}\ge0\forall x\)(vì \(x^2+9>0\forall x\)).

\(\Rightarrow-\frac{\left(2x+3\right)^2}{x^2+9}\le0\forall x\).

\(\Rightarrow4-\frac{\left(2x+3\right)^2}{x^2+9}\le4\forall x\).

\(\Rightarrow A\le4\).

Dấu bằng xảy ra.

\(\Leftrightarrow\left(2x+3\right)^2=0\Leftrightarrow x=-\frac{3}{2}\).

Vậy \(maxA=4\Leftrightarrow x=-\frac{3}{2}\).

a, GTLN của A = 6