(sqrt(2)+1)^((6*x-6)/(x+1))<=(sqrt(2)-1)^(-x) (неравенство)

Укажите решение неравенства: (sqrt(2)+1)^((6*x-6)/(x+1))<=(sqrt(2)-1)^(-x) (множество решений неравенства)

Вы ввели

           6*x - 6                 
           -------                 
            x + 1                -x
/  ___    \           /  ___    \  
\\/ 2  + 1/        <= \\/ 2  - 1/  

$$\left(1 + \sqrt{2}\right)^{\frac{6 x - 6}{x + 1}} \leq \left(-1 + \sqrt{2}\right)^{- x}$$

Подробное решение

Дано неравенство:
$$\left(1 + \sqrt{2}\right)^{\frac{6 x - 6}{x + 1}} \leq \left(-1 + \sqrt{2}\right)^{- x}$$
Чтобы решить это нер-во - надо сначала решить соотвествующее ур-ние:
$$\left(1 + \sqrt{2}\right)^{\frac{6 x - 6}{x + 1}} = \left(-1 + \sqrt{2}\right)^{- x}$$
Решаем:
$$x_{1} = - \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(\sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}} + \log{\left (- \left(1 + \sqrt{2}\right)^{6} \left(- \sqrt{2} + 1\right) \right )}\right)$$
$$x_{2} = \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(- \log{\left (\left(-1 + \sqrt{2}\right) \left(1 + \sqrt{2}\right)^{6} \right )} + \sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}}\right)$$
$$x_{1} = - \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(\sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}} + \log{\left (- \left(1 + \sqrt{2}\right)^{6} \left(- \sqrt{2} + 1\right) \right )}\right)$$
$$x_{2} = \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(- \log{\left (\left(-1 + \sqrt{2}\right) \left(1 + \sqrt{2}\right)^{6} \right )} + \sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}}\right)$$
Данные корни
$$x_{2} = \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(- \log{\left (\left(-1 + \sqrt{2}\right) \left(1 + \sqrt{2}\right)^{6} \right )} + \sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}}\right)$$
$$x_{1} = - \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(\sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}} + \log{\left (- \left(1 + \sqrt{2}\right)^{6} \left(- \sqrt{2} + 1\right) \right )}\right)$$
являются точками смены знака неравенства в решениях.
Сначала определимся со знаком до крайней левой точки:
$$x_{0} \leq x_{2}$$
Возьмём например точку
$$x_{0} = x_{2} - \frac{1}{10}$$
=

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  /       2/       ___\         2/      ___\      |/       ___\                  |       |/      ___\  /       ___\|     
\/     log \-1 + \/ 2 / + 36*log \1 + \/ 2 / + log\\-1 + \/ 2 /                  /  - log\\1 + \/ 2 / *\-1 + \/ 2 //   1 
-------------------------------------------------------------------------------------------------------------------- - --
                                                      1/       ___\                                                    10
                                                 2*log \-1 + \/ 2 /                                                      

=
$$- \frac{1}{10} + \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(- \log{\left (\left(-1 + \sqrt{2}\right) \left(1 + \sqrt{2}\right)^{6} \right )} + \sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}}\right)$$
подставляем в выражение
$$\left(1 + \sqrt{2}\right)^{\frac{6 x - 6}{x + 1}} \leq \left(-1 + \sqrt{2}\right)^{- x}$$

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             |\/     log \-1 + \/ 2 / + 36*log \1 + \/ 2 / + log\\-1 + \/ 2 /                  /  - log\\1 + \/ 2 / *\-1 + \/ 2 //   1 |                                                                                                                                               
           6*|-------------------------------------------------------------------------------------------------------------------- - --| - 6                                                                                                                                           
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            |\/     log \-1 + \/ 2 / + 36*log \1 + \/ 2 / + log\\-1 + \/ 2 /                  /  - log\\1 + \/ 2 / *\-1 + \/ 2 //   1     |                 |\/     log \-1 + \/ 2 / + 36*log \1 + \/ 2 / + log\\-1 + \/ 2 /                  /  - log\\1 + \/ 2 / *\-1 + \/ 2 //   1 |
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/  ___    \                                                                                                                                     /  ___    \                                                                                                                            
\\/ 2  + 1/                                                                                                                                  <= \\/ 2  - 1/                                                                                                                            

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                    |  /       2/       ___\         2/      ___\      |/       ___\                  |       |/      ___\  /       ___\||                                                                                                                                         
             33   3*\\/     log \-1 + \/ 2 / + 36*log \1 + \/ 2 / + log\\-1 + \/ 2 /                  /  - log\\1 + \/ 2 / *\-1 + \/ 2 ///                           _____________________________________________________________________________                                 
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                         _____________________________________________________________________________                                     <=             1    \/     log \-1 + \/ 2 / + 36*log \1 + \/ 2 / + log\\-1 + \/ 2 /                  /  - log\\1 + \/ 2 / *\-1 + \/ 2 //
                        /                                            /               /           36\\                                                     -- - --------------------------------------------------------------------------------------------------------------------
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                      /                                              |            log\\1 + \/ 2 /  /|       /           6             \                                                                         2*log\-1 + \/ 2 /                                                  
                     /       2/       ___\         2/      ___\      |/       ___\                  |       |/      ___\  /       ___\|       /       ___\                                                                                                                         
              9    \/     log \-1 + \/ 2 / + 36*log \1 + \/ 2 / + log\\-1 + \/ 2 /                  /  - log\\1 + \/ 2 / *\-1 + \/ 2 //       \-1 + \/ 2 /                                                                                                                         
              -- + --------------------------------------------------------------------------------------------------------------------                                                                                                                                            
              10                                                         /       ___\                                                                                                                                                                                              
                                                                    2*log\-1 + \/ 2 /                                                                                                                                                                                              
/      ___\                                                                                                                                                                                                                                                                        
\1 + \/ 2 /                                                                                                                                                                                                                                                                        

значит одно из решений нашего неравенства будет при:
$$x \leq \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(- \log{\left (\left(-1 + \sqrt{2}\right) \left(1 + \sqrt{2}\right)^{6} \right )} + \sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}}\right)$$

 _____           _____          
      \         /
-------•-------•-------
       x2      x1

Другие решения неравенства будем получать переходом на следующий полюс
и т.д.
Ответ:
$$x \leq \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(- \log{\left (\left(-1 + \sqrt{2}\right) \left(1 + \sqrt{2}\right)^{6} \right )} + \sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}}\right)$$
$$x \geq - \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(\sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}} + \log{\left (- \left(1 + \sqrt{2}\right)^{6} \left(- \sqrt{2} + 1\right) \right )}\right)$$

Быстрый ответ

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  |   |        /                                              |            log\\1 + \/ 2 /  /|       /           6             \        |     | |   /                                              |            log\\1 + \/ 2 /  /|       /            6            \|              ||
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  |   |     \/     log \-1 + \/ 2 / + 36*log \1 + \/ 2 / + log\\-1 + \/ 2 /                  /  - log\\1 + \/ 2 / *\-1 + \/ 2 //        |     |-\\/     log \-1 + \/ 2 / + 36*log \1 + \/ 2 / + log\\-1 + \/ 2 /                  /  + log\-\1 + \/ 2 / *\1 - \/ 2 ///              ||
Or|And|x <= --------------------------------------------------------------------------------------------------------------------, -1 < x|, And|------------------------------------------------------------------------------------------------------------------------ <= x, x < oo||
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  \   \                                                      2*log\-1 + \/ 2 /                                                          /     \                                                   2*log\-1 + \/ 2 /                                                                 //

$$\left(x \leq \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(- \log{\left (\left(-1 + \sqrt{2}\right) \left(1 + \sqrt{2}\right)^{6} \right )} + \sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}}\right) \wedge -1 < x\right) \vee \left(- \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(\sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}} + \log{\left (- \left(1 + \sqrt{2}\right)^{6} \left(- \sqrt{2} + 1\right) \right )}\right) \leq x \wedge x < \infty\right)$$

Быстрый ответ 2

           _____________________________________________________________________________                                       /      _____________________________________________________________________________                                 \      
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     \/     log \-1 + \/ 2 / + 36*log \1 + \/ 2 / + log\\-1 + \/ 2 /                  /  - log\\1 + \/ 2 / *\-1 + \/ 2 //     -\\/     log \-1 + \/ 2 / + 36*log \1 + \/ 2 / + log\\-1 + \/ 2 /                  /  + log\-\1 + \/ 2 / *\1 - \/ 2 ///      
(-1, --------------------------------------------------------------------------------------------------------------------] U [------------------------------------------------------------------------------------------------------------------------, oo)
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                                                      2*log\-1 + \/ 2 /                                                                                                          2*log\-1 + \/ 2 /                                                         

$$x \in \left(-1, \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(- \log{\left (\left(-1 + \sqrt{2}\right) \left(1 + \sqrt{2}\right)^{6} \right )} + \sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}}\right)\right] \cup \left[- \frac{1}{2 \log{\left (-1 + \sqrt{2} \right )}} \left(\sqrt{\log{\left (\left(-1 + \sqrt{2}\right)^{\log{\left (\left(1 + \sqrt{2}\right)^{36} \right )}} \right )} + \log^{2}{\left (-1 + \sqrt{2} \right )} + 36 \log^{2}{\left (1 + \sqrt{2} \right )}} + \log{\left (- \left(1 + \sqrt{2}\right)^{6} \left(- \sqrt{2} + 1\right) \right )}\right), \infty\right)$$