\[\boxed{\mathbf{1467}\mathbf{.}}\]
\[y = x^{2} - 2x - 3\]
\[1)\ y(x) < 0:\]
\[x^{2} - 2x - 3 < 0\]
\[D = 4 + 12 = 16\]
\[x_{1} = \frac{2 - 4}{2} = - 1;\]
\[x_{2} = \frac{2 + 4}{2} = 3;\]
\[(x + 1)(x - 3) < 0\]
\[- 1 < x < 3.\]
\[2)\ Возрастает\ на\ \lbrack 1;\ 4\rbrack:\]
\[y^{'}(x) = \left( x^{2} \right)^{'} - (2x + 3)^{'} =\]
\[= 2x - 2;\]
\[2x - 2 \geq 0\]
\[x - 1 \geq 0\]
\[x \geq 1.\]
\[Что\ и\ требовалось\ доказать.\]
\[3)\ Наименьшее\ значение\ \]
\[функции:\]
\[x = 1 - точка\ минимума;\]
\[y(1) = 1^{2} - 2 \bullet 1 - 3 =\]
\[= 1 - 2 - 3 = - 4.\]
\[4)\ Выше\ графика\ функции\ \]
\[y = - 2x + 1:\]
\[x^{2} - 2x - 3 > - 2x + 1\]
\[x^{2} > 4\]
\[x < - 2\ \ и\ \ x > 2.\]
\[5)\ Уравнение\ касательной\ \]
\[в\ точке\ x = 2:\]
\[f^{'}(x) = \left( x^{2} \right)^{'} - (2x + 3)^{'} =\]
\[= 2x - 2;\]
\[f^{'}(2) = 2 \bullet 2 - 2 = 4 - 2 = 2;\]
\[f(2) = 2^{2} - 2 \bullet 2 - 3 =\]
\[= 4 - 4 - 3 = - 3;\]
\[y = - 3 + 2(x - 2) =\]
\[= - 3 + 2x - 4 = 2x - 7.\]