14А. Найдите все значения параметра a, при каждом из которых корни уравнения \({x^2}-2a\,x + {a^2}-2 = 0\) принадлежат отрезку \(\left[ {2;\,5} \right]\).
\(\left\{ {\begin{array}{*{20}{c}}{D \ge 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{2 \le {x_B} = -\frac{b}{{2a}} \le 5,}\\{f\left( 2 \right) \ge 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{f\left( 5 \right) \ge 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{4{a^2}-4\left( {{a^2}-2} \right) \ge 0,\,\,}\\{2 \le a \le 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{4-4a + {a^2}-2 \ge 0,\,\,\,\,}\\{25-10a + {a^2}-2 \ge 0}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{8 \ge 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{2 \le a \le 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{a^2}-4a + 2 \ge 0,\,\,\,}\\{{a^2}-10a + 23 \ge 0}\end{array}} \right.\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{a \in R,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{a \in \left[ {2;\,5} \right],\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{a\, \in \,\left( {-\infty ;2-\sqrt 2 } \right] \cup \left[ {2 + \sqrt 2 ;\infty } \right),}\\{a\, \in \,\left( {-\infty ;5-\sqrt 2 } \right] \cup \left[ {5 + \sqrt 2 ;\infty } \right)\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,a\, \in \,\left[ {2 + \sqrt 2 ;5-\sqrt 2 } \right].\) Таким образом, корни уравнения будут принадлежать отрезку \(\left[ {2;\,5} \right]\) при \(a\, \in \,\left[ {2 + \sqrt 2 ;5-\sqrt 2 } \right].\) ОТВЕТ: \(\left[ {2 + \sqrt 2 ;\,5-\sqrt 2 } \right].\)
Введём функцию \(f\left( x \right) = f\left( x \right) = {x^2}-2ax + {a^2}-2\) графиком которой является парабола ветвями вверх. Для того чтобы корни уравнения принадлежали отрезку\(\left[ {2;\,5} \right]\), необходимо выполнение следующих условий (см. рис.):