Đáp án:

$a) max=2  \Leftrightarrow x \ge 5\\ b)max_D \Leftrightarrow x =-\dfrac{5}{2}\\ c)max_P =4\Leftrightarrow \left\{\begin{array}{l} x=\dfrac{2}{5} \\ y=-4\end{array} \right.\\ d)max_G = 5,5 \Leftrightarrow x =\dfrac{3}{4}\\ e) max_E=- 14,2 x =\dfrac{17}{5}.$

Giải thích các bước giải:

$a) |x-3| - |5-x|= |x-3| - |x-5| \le |x-3-(x-5)|=|2|=2$

Dấu "=" xảy ra $\Leftrightarrow  \left\{\begin{array}{l} |x-3| \ge |x-5| \\ (x-3)(x-5) \ge 0\end{array} \right.$

$\Leftrightarrow  \left\{\begin{array}{l} |x-3| \ge |x-5| \\ \left[\begin{array}{l} \left\{\begin{array}{l} x-3 \ge 0 \\ x-5 \ge 0\end{array} \right. \\  \left\{\begin{array}{l} x-3 \le 0 \\ x-5 \le 0 \end{array} \right. \end{array} \right. \end{array} \right.\\ \Leftrightarrow  \left\{\begin{array}{l} |x-3| \ge |x-5| \\ \left[\begin{array}{l}  \left\{\begin{array}{l} x \ge 3 \\ x \ge 5 \end{array} \right. \\  \left\{\begin{array}{l} x \le 3 \\ x \le 5 \end{array} \right. \end{array} \right. \end{array} \right. \\ \Leftrightarrow  \left\{\begin{array}{l} |x-3| \ge |x-5| \\  \left[\begin{array}{l} x \ge 5 \\ x \le 3 \end{array} \right. \end{array} \right.\\ \Leftrightarrow  \left[\begin{array}{l}  \left\{\begin{array}{l} |x-3| \ge |x-5| \\ x \ge 5 \end{array} \right. \\  \left\{\begin{array}{l} |x-3| \ge |x-5|\\ x \le 3 \end{array} \right. \end{array} \right.\\ \Leftrightarrow  \left[\begin{array}{l}  \left\{\begin{array}{l} x-3 \ge x-5 \\ x \ge 5 \end{array} \right. \\  \left\{\begin{array}{l} -(x-3) \ge -(x-5)\\ x \le 3 \end{array} \right. \end{array} \right.\\ \Leftrightarrow  \left[\begin{array}{l}  \left\{\begin{array}{l} -3 \ge -5 \\ x \ge 5 \end{array} \right. \\  \left\{\begin{array}{l} 3 \ge 5(\text{Vô lí)}\\ x \le 3 \end{array} \right. \end{array} \right.\\ \Leftrightarrow x \ge 5$

$b)D = -\left|x +\dfrac{5}{2} \right| \le 0 \ \forall \ x$

Dấu "=" xảy ra $\Leftrightarrow x +\dfrac{5}{2}=0 \Leftrightarrow x =-\dfrac{5}{2}$

$c)P =4 - |5x-2| - |3y + 12| \le 4 \ \forall \ x,y$

Dấu "=" xảy ra $\Leftrightarrow \left\{\begin{array}{l}5x-2 =0 \\3y + 12=0 \end{array} \right.\Leftrightarrow \left\{\begin{array}{l} x=\dfrac{2}{5} \\ y=-4\end{array} \right.$

$d)G = 5,5 - |2x - 1,5| \le 5,5 \ \forall \ x$

Dấu "=" xảy ra $\Leftrightarrow 2x - 1,5=0 \Leftrightarrow x =\dfrac{3}{4}$

$e) -|10,2 - 3x| - 14,2 \le - 14,2 \ \forall \ x$

Dấu "=" xảy ra $\Leftrightarrow 10,2 - 3x=0 \Leftrightarrow x =\dfrac{17}{5}.$