Đáp án:
$a)max=3 \Leftrightarrow x=1\\ b)max= 33 \Leftrightarrow x=-4\\ c) max=\dfrac{109}{4} \Leftrightarrow x=\dfrac{7}{2}\\ d)max=\dfrac{69}{4} \Leftrightarrow x=-\dfrac{5}{2}\\ e)max=87 \Leftrightarrow\left\{\begin{array}{l} x=2\\ y=-6\end{array} \right.\\ g) max=\dfrac{31}{2} \Leftrightarrow\left\{\begin{array}{l} x=-\dfrac{1}{2}\\ y=-\dfrac{3}{2}\end{array} \right.$
Giải thích các bước giải:
$a) -x^2+2x+2\\ = -x^2+2x-1+3\\ =-(x-1)^2+3 \le 3 \ \forall \ x$
Dấu "=" xảy ra $\Leftrightarrow x-1=0 \Leftrightarrow x=1$
$b) -x^2-8x+17\\ =-x^2-8x-16+33\\ =-(x+4)^2+33 \le 33 \ \forall \ x$
Dấu "=" xảy ra $\Leftrightarrow x+4=0 \Leftrightarrow x=-4$
$c) -x^2+7x+15\\ =-x^2+2.\dfrac{7}{2}x-\dfrac{49}{4}+\dfrac{109}{4}\\ =-\left(x-\dfrac{7}{2}\right)^2+\dfrac{109}{4} \le \dfrac{109}{4} \ \forall \ x$
Dấu "=" xảy ra $\Leftrightarrow x-\dfrac{7}{2}=0 \Leftrightarrow x=\dfrac{7}{2}$
$d) -x^2-5x+11\\ = -x^2-2.\dfrac{5}{2}x-\dfrac{25}{4}+\dfrac{69}{4}\\ =-\left(x+\dfrac{5}{2}\right)^2+\dfrac{69}{4} \le \dfrac{69}{4} \ \forall \ x$
Dấu "=" xảy ra $\Leftrightarrow x+\dfrac{5}{2}=0 \Leftrightarrow x=-\dfrac{5}{2}$
$e)-x^2+4x-y^2-12y+47\\ =-x^2+4x-4-y^2-12y-36+87\\ =-(x-2)^2-(y+6)^2+87 \le 87 \ \forall \ x,y$
Dấu "=" xảy ra $\Leftrightarrow \left\{\begin{array}{l} x-2=0\\ y+6=0\end{array} \right. \Leftrightarrow \left\{\begin{array}{l} x=2\\ y=-6\end{array} \right.$
$g) -x^2-x-y^2-3y+13\\ =-x^2-x-\dfrac{1}{4}-y^2-2.\dfrac{3}{2}y-\dfrac{9}{4}+\dfrac{31}{2}\\ =-\left(x+\dfrac{1}{2}\right)^2-\left(y+\dfrac{3}{2}\right)^2+\dfrac{31}{2} \le \dfrac{31}{2} \ \forall \ x,y$
Dấu "=" xảy ra $\Leftrightarrow \left\{\begin{array}{l} x+\dfrac{1}{2}=0\\ y+\dfrac{3}{2}=0\end{array} \right. \Leftrightarrow \left\{\begin{array}{l} x=-\dfrac{1}{2}\\ y=-\dfrac{3}{2}\end{array} \right.$