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МатематикаРешебник по алгебре и начала математического анализа 11 класс Алимов Задание 927
Задание 927
\[\boxed{\mathbf{927}\mathbf{.}}\]
\[1)\ y = - x^{4} + 8x^{2} - 16\]
\[\textbf{а)}\ D(x) = ( - \infty;\ + \infty);\]
\[\textbf{б)}\ y^{'}(x) =\]
\[= - \left( x^{4} \right)^{'} + 8 \bullet \left( x^{2} \right)^{'} - (16)^{'};\]
\[y^{'}(x) = - 4x^{3} + 8 \bullet 2x - 0 =\]
\[= - 4x^{3} + 16x;\]
\[\textbf{в)}\ Стационарные\ точки:\]
\[16x - 4x^{3} = 0\]
\[4x \bullet \left( 4 - x^{2} \right) = 0\]
\[(2 + x) \bullet 4x \bullet (2 - x) = 0\]
\[x_{1} = - 2,\ \ \ x_{2} = 0,\ \ \ x_{3} = 2.\]
\[\textbf{г)}\ f( \pm 2) =\]
\[= - ( \pm 2)^{4} + 8 \bullet ( \pm 2)^{2} - 16 =\]
\[= - 16 + 32 - 16 = 0;\]
\[f(0) = - 0^{4} + 8 \bullet 0^{2} - 16 =\]
\[= - 16.\]
\[\textbf{д)}\ Возрастает\ на\ \]
\[( - \infty;\ - 2) \cup (0;\ 2)\ и\ убывает\ \]
\[на\ ( - 2;\ 0) \cup (2;\ + \infty);\]
\[x = 0 - точка\ минимума;\ \ \]
\[x = \pm 2 - точки\ максимума.\]
\[\textbf{е)}\ \]
| \[x\] | \[x < - 2\] | \[- 2\] | \[- 2 < x < 0\] | \[0\] | \[0 < x < 2\] | \[2\] | \[x > 2\] |
|---|---|---|---|---|---|---|---|
| \[f^{'}(x)\] | \[+\] | \[0\] | \[-\] | \[0\] | \[+\] | \[0\] | \[-\] |
| \[f(x)\] | \[\nearrow\] | \[0\] | \[\searrow\] | \[- 16\] | \[\nearrow\] | \[0\] | \[\searrow\] |
\[2)\ y = x^{4} - 2x^{2} + 2\]
\[\textbf{а)}\ D(x) = ( - \infty;\ + \infty);\]
\[\textbf{б)}\ y^{'}(x) =\]
\[= \left( x^{4} \right)^{'} - 2 \bullet \left( x^{2} \right)^{'} + (2)';\]
\[y^{'}(x) = 4x^{3} - 2 \bullet 2x + 0 =\]
\[= 4x^{3} - 4x;\]
\[\textbf{в)}\ Стационарные\ точки:\]
\[4x \bullet \left( x^{2} - 1 \right) = 0\]
\[(x + 1) \bullet 4x \bullet (x - 1) = 0\]
\[x_{1} = - 1,\ \ \ x_{2} = 0,\ \ \ x_{3} = 1.\]
\[\textbf{г)}\ f( \pm 1) =\]
\[= ( \pm 1)^{4} - 2 \bullet ( \pm 1)^{2} + 2 =\]
\[= 1 - 2 + 2 = 1;\]
\[f(0) = 0^{4} - 2 \bullet 0^{2} + 2 = 2.\]
\[\textbf{д)}\ Возрастает\ \]
\[на\ ( - 1;\ 0) \cup (1;\ + \infty)\ и\ убывает\ \]
\[на\ ( - \infty;\ - 1) \cup (0;\ 1);\]
\[x = \pm 1 - точки\ минимума;\ \ \]
\[x = 0 - точка\ максимума.\]
\[\textbf{е)}\ \]
| \[x\] | \[x < - 1\] | \[- 1\] | \[- 1 < x < 0\] | \[0\] | \[0 < x < 1\] | \[1\] | \[x > 1\] |
|---|---|---|---|---|---|---|---|
| \[f^{'}(x)\] | \[-\] | \[0\] | \[+\] | \[0\] | \[-\] | \[0\] | \[+\] |
| \[f(x)\] | \[\searrow\] | \[1\] | \[\nearrow\] | \[2\] | \[\searrow\] | \[1\] | \[\nearrow\] |
\[3)\ y = \frac{1}{4}x^{4} - \frac{1}{24}x^{6}\]
\[\textbf{а)}\ D(x) = ( - \infty;\ + \infty);\]
\[\textbf{б)}\ y^{'}(x) = \frac{1}{4} \bullet \left( x^{4} \right)^{'} - \frac{1}{24} \bullet \left( x^{6} \right)^{'};\]
\[y^{'}(x) = \frac{1}{4} \bullet 4x^{3} - \frac{1}{24} \bullet 6x^{5} =\]
\[= x^{3} - \frac{1}{4}x^{5}.\]
\[\textbf{в)}\ Стационарные\ точки:\]
\[x^{3} - \frac{1}{4}x^{5} = 0\]
\[x^{3} \bullet \left( 1 - 0,25x^{2} \right) = 0\]
\[(1 + 0,5x) \bullet x \bullet (1 - 0,5x) = 0\]
\[x_{1} = - 2,\ \ \ x_{2} = 0,\ \ \ x_{3} = 2.\]
\[\textbf{г)}\ f( \pm 2) =\]
\[= \frac{1}{4} \bullet ( \pm 2)^{4} - \frac{1}{24} \bullet ( \pm 2)^{6} =\]
\[= \frac{16}{4} - \frac{64}{24} = \frac{4}{3} = 1\frac{1}{3};\]
\[f(0) = \frac{1}{4} \bullet 0^{4} - \frac{1}{24} \bullet 0^{6} = 0.\]
\[\textbf{д)}\ Возрастает\ \]
\[на\ ( - \infty;\ - 2) \cup (0;\ 2)\ и\ убывает\ \]
\[на\ ( - 2;\ 0) \cup (2;\ + \infty);\]
\[x = 0 - точка\ минимума;\ \ \]
\[x = \pm 2 - точки\ максимума.\]
\[\textbf{е)}\ \]
| \[x\] | \[x < - 2\] | \[- 2\] | \[- 2 < x < 0\] | \[0\] | \[0 < x < 2\] | \[2\] | \[x > 2\] |
|---|---|---|---|---|---|---|---|
| \[f^{'}(x)\] | \[+\] | \[0\] | \[-\] | \[0\] | \[+\] | \[0\] | \[-\] |
| \[f(x)\] | \[\nearrow\] | \[1\frac{1}{3}\] | \[\searrow\] | \[0\] | \[\nearrow\] | \[1\frac{1}{3}\] | \[\searrow\] |
\[4)\ y = 6x^{4} - 4x^{6}\]
\[\textbf{а)}\ D(x) = ( - \infty;\ + \infty);\]
\[\textbf{б)}\ y^{'}(x) = 6 \bullet \left( x^{4} \right)^{'} - 4 \bullet \left( x^{6} \right)^{'};\]
\[y^{'}(x) = 6 \bullet 4x^{3} - 4 \bullet 6x^{5} =\]
\[= 24x^{3} - 24x^{5}.\]
\[\textbf{в)}\ Стационарные\ точки:\]
\[24x^{3} - 24x^{5} = 0\]
\[24x^{3} \bullet \left( 1 - x^{2} \right) = 0\]
\[(1 + x) \bullet x \bullet (1 - x) = 0\]
\[x_{1} = - 1,\ \ \ x_{2} = 0,\ \ \ x_{3} = 1.\]
\[\textbf{г)}\ f( \pm 1) =\]
\[= 6 \bullet ( \pm 1)^{4} - 4 \bullet ( \pm 1)^{6} =\]
\[= 6 - 4 = 2;\]
\[f(0) = 6 \bullet 0^{4} - 4 \bullet 0^{6} = 0.\]
\[\textbf{д)}\ Возрастает\ на\ \]
\[( - \infty;\ - 1) \cup (0;\ 1)\ и\ убывает\ \]
\[на\ ( - 1;\ 0) \cup (1;\ + \infty);\]
\[x = 0 - точка\ минимума;\ \ \]
\[x = \pm 1 - точки\ максимума.\]
\[\textbf{е)}\ \]
| \[x\] | \[x < - 1\] | \[- 1\] | \[- 1 < x < 0\] | \[0\] | \[0 < x < 1\] | \[1\] | \[x > 1\] |
|---|---|---|---|---|---|---|---|
| \[f^{'}(x)\] | \[+\] | \[0\] | \[-\] | \[0\] | \[+\] | \[0\] | \[-\] |
| \[f(x)\] | \[\nearrow\] | \[2\] | \[\searrow\] | \[0\] | \[\nearrow\] | \[2\] | \[\searrow\] |
