\(A=19-3x^2-24x\)
\(\Leftrightarrow A=-3\left(x^2+8x-\dfrac{19}{3}\right)\)
\(\Leftrightarrow A=-3\left(x^2+2x.4+16-\dfrac{67}{3}\right)\)
\(\Leftrightarrow A=-3\left[\left(x+4\right)^2-\dfrac{67}{3}\right]\)
\(\Leftrightarrow A=-3\left(x+4\right)^2+67\le67\forall x\)
Dấu " = " xảy ra
\(\Leftrightarrow-3\left(x+4\right)^2=0\Leftrightarrow\left(x+4\right)^2=0\Leftrightarrow x+4=0\Leftrightarrow x=-4\)
Vậy Max A là : 67 \(\Leftrightarrow x=-4\)
\(B=-x^2+6x-23\)
\(\Leftrightarrow B=-\left(x^2-6x+9\right)-14\)
\(\Leftrightarrow B=-\left(x-3\right)^2-14\le-14\forall x\)
Dấu " = " xảy ra
\(\Leftrightarrow-\left(x-3\right)^2=0\Leftrightarrow\left(x-3\right)^2=0\Leftrightarrow x-3=0\Leftrightarrow x=3\)
Vậy Max B là : \(-14\Leftrightarrow x=3\)
\(C=4\left(x-1\right)^2-9\left(x+2\right)^2\)
\(\Leftrightarrow C=4x^2-8x+4-9x^2-36x-36\)
\(\Leftrightarrow C=-5x^2-44x-32\)
\(\Leftrightarrow C=-5\left(x^2+\dfrac{44}{5}x+\dfrac{32}{5}\right)\)
\(\Leftrightarrow C=-5\left(x^2+2x.\dfrac{22}{5}+\dfrac{484}{25}\right)+64,8\)
\(\Leftrightarrow C=-5\left(x+\dfrac{22}{5}\right)^2+64,8\le64,8\forall x\)
Dấu " = " xảy ra
\(\Leftrightarrow-5\left(x+\dfrac{22}{5}\right)^2=0\Leftrightarrow\left(x+\dfrac{22}{5}\right)^2=0\Leftrightarrow x+\dfrac{22}{5}=0\)
\(\Leftrightarrow x=-\dfrac{22}{5}\)
Vậy Max C là : 64 , 8 \(\Leftrightarrow x=-\dfrac{22}{5}\)
\(E=\left(x+2\right)^2-2x^2+8\)
\(\Leftrightarrow E=x^2+4x+4-2x^2+8\)
\(\Leftrightarrow E=-x^2+4x+12\)
\(\Leftrightarrow E=-\left(x^2-4x+4\right)+16\)
\(\Leftrightarrow E=-\left(x-2\right)^2+16\le16\forall x\)
Dấu " = " xảy ra
\(\Leftrightarrow-\left(x-2\right)^2=0\Leftrightarrow\left(x-2\right)^2=0\Leftrightarrow x-2=0\Leftrightarrow x=2\)
Vậy Max E là : \(16\Leftrightarrow x=2\)
Z