1a)
![8^x+\frac{64}{8^x}=65, \\ 8^{2x}-65\cdot8^x+64=0, \\ 8^x=t, \\ t^2-65t+64=0, \\ t_1=1, t_2=64, \\ 8^x=1, 8^x=8^0, x_1=0, \\ 8^x=64, 8^x=8^2, x_2=2; 8^x+\frac{64}{8^x}=65, \\ 8^{2x}-65\cdot8^x+64=0, \\ 8^x=t, \\ t^2-65t+64=0, \\ t_1=1, t_2=64, \\ 8^x=1, 8^x=8^0, x_1=0, \\ 8^x=64, 8^x=8^2, x_2=2;](https://tex.z-dn.net/?f=8%5Ex%2B%5Cfrac%7B64%7D%7B8%5Ex%7D%3D65%2C+%5C%5C+8%5E%7B2x%7D-65%5Ccdot8%5Ex%2B64%3D0%2C+%5C%5C+8%5Ex%3Dt%2C+%5C%5C+t%5E2-65t%2B64%3D0%2C+%5C%5C+t_1%3D1%2C+t_2%3D64%2C+%5C%5C+8%5Ex%3D1%2C+8%5Ex%3D8%5E0%2C+x_1%3D0%2C+%5C%5C+8%5Ex%3D64%2C+8%5Ex%3D8%5E2%2C+x_2%3D2%3B)
1b)
1, f'(x)>0, f(x)\searrow, \\ x_{min}=1, f(1)=8^1+\frac{64}{8^1}=16;" alt="f'(x)=8^x\ln8-\frac{64}{8^{2x}}\cdot8^x\ln8=8^x\ln8-\frac{64\ln8}{8^x}, \\ f'(x)=0, \ 8^x\ln8-\frac{64\ln8}{8^x}=0, \\ \frac{\ln8}{8^x}(8^{2x}-64)=0, \\ 8^{2x}-64=0, \\ 8^{2x}=8^2, \\ 2x=2, \\ x=1, \\ x<1, f'(x)<0, f(x)\nearrow, \\ x>1, f'(x)>0, f(x)\searrow, \\ x_{min}=1, f(1)=8^1+\frac{64}{8^1}=16;" align="absmiddle" class="latex-formula">
1c)
![f(1-x)=f(1+x), f(1-x)-f(1+x)=0, \\ f(1-x)-f(1+x)=8^{1-x}+\frac{64}{8^{1-x}}-(8^{1+x}+\frac{64}{8^{1+x}})= \\ =\frac{8}{8^x}+\frac{64\cdot8^x}{8}-8\cdot8^x-\frac{64}{8\cdot8^x}=\frac{8}{8^x}+8\cdot8^x-8\cdot8^x-\frac{8}{8^x}=0; f(1-x)=f(1+x), f(1-x)-f(1+x)=0, \\ f(1-x)-f(1+x)=8^{1-x}+\frac{64}{8^{1-x}}-(8^{1+x}+\frac{64}{8^{1+x}})= \\ =\frac{8}{8^x}+\frac{64\cdot8^x}{8}-8\cdot8^x-\frac{64}{8\cdot8^x}=\frac{8}{8^x}+8\cdot8^x-8\cdot8^x-\frac{8}{8^x}=0;](https://tex.z-dn.net/?f=f%281-x%29%3Df%281%2Bx%29%2C+f%281-x%29-f%281%2Bx%29%3D0%2C+%5C%5C+f%281-x%29-f%281%2Bx%29%3D8%5E%7B1-x%7D%2B%5Cfrac%7B64%7D%7B8%5E%7B1-x%7D%7D-%288%5E%7B1%2Bx%7D%2B%5Cfrac%7B64%7D%7B8%5E%7B1%2Bx%7D%7D%29%3D+%5C%5C+%3D%5Cfrac%7B8%7D%7B8%5Ex%7D%2B%5Cfrac%7B64%5Ccdot8%5Ex%7D%7B8%7D-8%5Ccdot8%5Ex-%5Cfrac%7B64%7D%7B8%5Ccdot8%5Ex%7D%3D%5Cfrac%7B8%7D%7B8%5Ex%7D%2B8%5Ccdot8%5Ex-8%5Ccdot8%5Ex-%5Cfrac%7B8%7D%7B8%5Ex%7D%3D0%3B)
1d)
График симметричен относительно х=1;
2a)
0, x<1, \\ 3x+17>0, x>-\frac{17}{3}, \\ -5\frac{2}{3}0, x<1, \\ 3x+17>0, x>-\frac{17}{3}, \\ -5\frac{2}{3}