Решебник по алгебре 11 класс Никольский Параграф 5. Применение производной Задание 66

\[\boxed{\mathbf{66}\mathbf{.}}\]

\[\textbf{а)}\ f(x) = \frac{1}{3}x^{3} - x^{2};\]

\[f^{'}(x) = \frac{1}{3} \cdot 3x^{2} - 2x = x^{2} - 2x;\]

\[f"(x) = 2x - 2.\]

\[\textbf{б)}\ f(x) = \frac{1}{3}x^{3} + \frac{1}{2}x^{2};\]

\[f^{'}(x) = \frac{1}{3} \cdot 3x^{2} + \frac{1}{2} \cdot 2x = x^{2} + x;\]

\[f"(x) = 2x + 1.\]

\[\textbf{в)}\ f(x) = 5x^{3} - 4x^{2} + 7x - 13\]

\[f^{'}(x) = 5 \cdot 3x^{2} - 4 \cdot 2x + 7 = 15x^{2} - 8x + 7;\]

\[f"(x) = 15 \cdot 2x - 8 = 30x - 8.\]

\[\textbf{г)}\ f(x) = - 13x^{5} + 4x^{3} - x;\]

\[f^{'}(x) = - 13 \cdot 5x^{4} + 4 \cdot 3x^{2} -\]

\[- 1 = - 65x^{4} + 12x^{2} - 1;\]

\[f"(x) = - 65 \cdot 4x^{3}\ + 12 \cdot 2x =\]

\[= - 260x³ + 24x.\]