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МатематикаРешебник по алгебре 11 класс Никольский Параграф 4. Производная Задание 22
Задание 22
\[\boxed{\mathbf{22}\mathbf{.}}\]
\[\textbf{а)}\ f'(x) = 6x\]
\[f^{'}(x) = \left( Ax^{n} + C \right)^{'} =\]
\[= a \cdot n \cdot x^{n - 1}\]
\[6x = A \cdot n \cdot x^{n - 1}\]
\[n - 1 = 1\]
\[n = 2.\]
\[6x = A \cdot 2x\]
\[A = 3.\]
\[f(x) = 3x^{2} + C;\ \ C - любое\]
\[\ число:\]
\[f(x) = 3x^{2} - 10.\]
\[\textbf{б)}\ f^{'}(x) = x^{2} - 1\]
\[f(x) = \left( Ax^{n} + Bx^{m} + C \right)^{'} =\]
\[= Anx^{n - 1} + Bmx^{m - 1}\]
\[x^{2} - 1 = Anx^{n - 1} + Bmx^{m - 1}\]
\[n - 1 = 2\]
\[n = 3;\ \]
\[A = \frac{1}{3}.\]
\[m - 1 = 0\]
\[m = 1;\]
\[B = - 1.\]
\[f(x) = \frac{1}{3}x^{3} + x + C;\ \ \]
\[C - любое\ число:\]
\[f(x) = \frac{1}{3}x^{2} - x + 5.\]
\[\textbf{в)}\ f^{'}(x) = 3x^{2} + 2x - 5\]
\[f(x) = \left( Ax^{n} + Bx^{m} + Dx^{k} + C \right)^{'} =\]
\[= Anx^{n - 1} + Bmx^{m - 1} + Dkx^{k - 1}\]
\[3x^{2} + 2x - 5 = Anx^{n - 1} +\]
\[+ Bmx^{m - 1} + Dkx^{k - 1}\]
\[n - 1 = 2\]
\[n = 3;\]
\[A = 1.\]
\[m - 1 = 1\]
\[m = 2;\]
\[B = 1.\]
\[k - 1 = 0\]
\[k = 1;\]
\[D = - 5.\]
\[f(x) = x^{3} + x^{2} - 5x + C;\ \]
\[\ C - любое\ число:\]
\[f(x) = x^{3} + x^{2} - 5x - 1.\]
\[\textbf{г)}\ f^{'}(x) = 6x^{2} - 4x + 7\]
\[f(x) = \left( Ax^{n} + Bx^{m} + Dx^{k} + C \right)^{'} =\]
\[= Anx^{n - 1} + Bmx^{m - 1} + Dkx^{k - 1}\]
\[6x^{2} - 4x + 7 = Anx^{n - 1} +\]
\[+ Bmx^{m - 1} + Dkx^{k - 1}\]
\[n - 1 = 2\]
\[n = 3;\]
\[A = 2.\]
\[m - 1 = 1\]
\[m = 2;\]
\[B = - 2.\]
\[k - 1 = 0\]
\[k = 1;\]
\[D = 7.\]
\[f(x) = 2x^{3} - 2x^{2} + 7x + C;\ \ \]
\[C - любое\ число:\]
\[f(x) = 2x^{3} - 2x^{2} + 7x - 5.\]
