Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 342

\[1)\ V(x) = (a - 2x)^{2} \bullet x =\]

\[= x\left( a^{2} - 4ax + 4x^{2} \right) =\]

\[= a^{2}x - 4ax^{2} + 4x^{3};\]

\[V^{'}(x) = a^{2} - 4a \bullet 2x + 4 \bullet 3x^{2} =\]

\[= a^{2} - 8ax + 12x^{2}.\]

\[2)\ a^{2} - 8ax + 12x^{2} \geq 0\]

\[D = 64x^{2} - 48x^{2} = 16x^{2}\]

\[a_{1} = \frac{8x - 4x}{2} = 2x;\text{\ \ }\]

\[a_{2} = \frac{8x + 4x}{2} = 6x;\]

\[(a - 2x)(a - 6x) \geq 0\]

\[(6x - a)(2x - a) \geq 0\]

\[x \leq \frac{a}{6};\ \ \ x \geq \frac{a}{2}.\]

\[3)\ Точка\ максимума:\]

\[x = \frac{a}{6}.\]

\[Ответ:\ \ \frac{a}{6}.\]