![f(x) = 9^x + 5*3^{-2x} = 3^{2x} + 5*3^{-2x}\\\\ f'(x) = 2*3^{2x}ln3 -10*3^{-2x}ln3\\\\](https://tex.z-dn.net/?f=f%28x%29+%3D+9%5Ex+%2B+5%2A3%5E%7B-2x%7D+%3D+3%5E%7B2x%7D+%2B+5%2A3%5E%7B-2x%7D%5C%5C%5C%5C+f%27%28x%29+%3D+2%2A3%5E%7B2x%7Dln3+-10%2A3%5E%7B-2x%7Dln3%5C%5C%5C%5C)
![f'(x) = 0; \ 2*3^{2x}ln3 -10*3^{-2x}ln3 = 0\\\\3^{2x} -5*3^{-2x} = 0\\\\ 3^{2x} = 5*3^{-2x}\\\\ 1/5 = 3^{-4x}, \ 3^{4x} = 5, \ 81^x = 5, \ x = log_{81}5\\](https://tex.z-dn.net/?f=f%27%28x%29+%3D+0%3B+%5C+2%2A3%5E%7B2x%7Dln3+-10%2A3%5E%7B-2x%7Dln3+%3D+0%5C%5C%5C%5C3%5E%7B2x%7D+-5%2A3%5E%7B-2x%7D+%3D+0%5C%5C%5C%5C+3%5E%7B2x%7D+%3D+5%2A3%5E%7B-2x%7D%5C%5C%5C%5C+1%2F5+%3D+3%5E%7B-4x%7D%2C+%5C+3%5E%7B4x%7D+%3D+5%2C+%5C+81%5Ex+%3D+5%2C+%5C+x+%3D+log_%7B81%7D5%5C%5C)
![f(x) < 0, x < log_{81}5,\\\\ f(x) > 0, x > log_{81}5\\\\](https://tex.z-dn.net/?f=f%28x%29+%3C+0%2C+x+%3C+log_%7B81%7D5%2C%5C%5C%5C%5C+f%28x%29+%3E+0%2C+x+%3E+log_%7B81%7D5%5C%5C%5C%5C)
Функция f(x) имеет в точке x = log_{81}5, минимум.
![f(log_{81}5) = 3^{2log_{81}5} + 5*3^{-2log_{81}5} = 3^{2log_{3^4}5} + 5*3^{-2log_{3^4}5} =\\\\ 3^{log_{3}\sqrt{5}} + 5*3^{log_{3}(1/\sqrt{5})} = \sqrt{5} + 5/\sqrt{5} = 2\sqrt{5}](https://tex.z-dn.net/?f=f%28log_%7B81%7D5%29+%3D+3%5E%7B2log_%7B81%7D5%7D+%2B+5%2A3%5E%7B-2log_%7B81%7D5%7D+%3D+3%5E%7B2log_%7B3%5E4%7D5%7D+%2B+5%2A3%5E%7B-2log_%7B3%5E4%7D5%7D+%3D%5C%5C%5C%5C+3%5E%7Blog_%7B3%7D%5Csqrt%7B5%7D%7D+%2B+5%2A3%5E%7Blog_%7B3%7D%281%2F%5Csqrt%7B5%7D%29%7D+%3D+%5Csqrt%7B5%7D+%2B+5%2F%5Csqrt%7B5%7D+%3D+2%5Csqrt%7B5%7D)
При x -> +∞, f(x) - > +∞, при x -> -∞, f(x) - > +∞
Mножество значений функции f(x): ![[2\sqrt{5}, +\infty)](https://tex.z-dn.net/?f=%5B2%5Csqrt%7B5%7D%2C+%2B%5Cinfty%29)
Решение:
ОДЗ:
![\left \{ {{2x\ \textgreater \ 0} \atop {x+1\ \textgreater \ 0}} \right. \\ \left \{ {{x\ \textgreater \ 0} \atop {x\ \textgreater \ -1}} \right.](https://tex.z-dn.net/?f=+%5Cleft+%5C%7B+%7B%7B2x%5C+%5Ctextgreater+%5C+0%7D+%5Catop+%7Bx%2B1%5C+%5Ctextgreater+%5C+0%7D%7D+%5Cright.+%5C%5C%0A+%5Cleft+%5C%7B+%7B%7Bx%5C+%5Ctextgreater+%5C+0%7D+%5Catop+%7Bx%5C+%5Ctextgreater+%5C+-1%7D%7D+%5Cright.+)
С учетом ОДЗ можем решить уравнение, раскрыв логарифм.
![2x = x+1 \\ x = 1](https://tex.z-dn.net/?f=2x+%3D+x%2B1+%5C%5C%0Ax+%3D+1)
Корень удовлетворяет ОДЗ, это и есть ответ.
Ответ: x = 1.
x^3 + 3x^2 - 4x - 12 = 0
x^3 + 3x^2 - 4x - 12 = (x^2 - 4)(x+3)
(x^2 - 4)(x+3)=0
x=2, x=-2, x=-3
<span>постройте график функции y=-x^2</span>
= 5 - 4 = 1. По формуле разности квадратов задача