25x^2=<0<br />x=<0<br />Ответ: (-~;0]
![\frac{3x-1}{6} - \frac{x}{3} = \frac{5-x}{9}](https://tex.z-dn.net/?f=+%5Cfrac%7B3x-1%7D%7B6%7D+-+%5Cfrac%7Bx%7D%7B3%7D+%3D+%5Cfrac%7B5-x%7D%7B9%7D+)
Приведем к общему знаменателю:
![\frac{3*(3x-1)}{3*6} - \frac{6*x}{6*3} = \frac{2*(5-x)}{2*9}](https://tex.z-dn.net/?f=%5Cfrac%7B3%2A%283x-1%29%7D%7B3%2A6%7D+-+%5Cfrac%7B6%2Ax%7D%7B6%2A3%7D+%3D+%5Cfrac%7B2%2A%285-x%29%7D%7B2%2A9%7D)
Избавимся от знаменателя ( умножим на 18):
3(3х-1) - 6х = 2 *(5-х)
9х-3-6х=10-2х
3х-3=10-2х
3х+2х=10+3
х=13/5 =2,6
Проверим:
![\frac{3*2,6-1}{6} - \frac{2,6}{3} = \frac{5-2,6}{9} \\ \\ \frac{6,8}{6} - \frac{2,6}{3} = \frac{2,4}{9} \\ \\ ](https://tex.z-dn.net/?f=%5Cfrac%7B3%2A2%2C6-1%7D%7B6%7D+-+%5Cfrac%7B2%2C6%7D%7B3%7D+%3D+%5Cfrac%7B5-2%2C6%7D%7B9%7D+%5C%5C++%5C%5C++%5Cfrac%7B6%2C8%7D%7B6%7D+-+%5Cfrac%7B2%2C6%7D%7B3%7D+%3D+%5Cfrac%7B2%2C4%7D%7B9%7D++%5C%5C++%5C%5C++%0A)
![\frac{20,4 -15,6}{18} = \frac{4,8}{18} \\ \\ \frac{4,8}{18} = \frac{4,8}{18} ](https://tex.z-dn.net/?f=%5Cfrac%7B20%2C4+-15%2C6%7D%7B18%7D+%3D+%5Cfrac%7B4%2C8%7D%7B18%7D++%5C%5C++%5C%5C++%5Cfrac%7B4%2C8%7D%7B18%7D+%3D+%5Cfrac%7B4%2C8%7D%7B18%7D+%0A)
Общий вид уравнения касательной:
.
Найдем значение функции в точке
, получим
![f(1)=3-\sqrt{1}-\frac{2}{\pi}\sin\pi =3-1-0=2](https://tex.z-dn.net/?f=f%281%29%3D3-%5Csqrt%7B1%7D-%5Cfrac%7B2%7D%7B%5Cpi%7D%5Csin%5Cpi%20%3D3-1-0%3D2)
Найдем производную функции
![f'(x)=(3-\sqrt{x}-\frac{2}{\pi}\sin \pi x)=(3)'-(\sqrt{x})'-\frac{2}{\pi}\cdot(\sin\pi x)'=\\ \\ =-\frac{1}{2\sqrt{x}}-\frac{2}{\pi}\cdot \cos\pi x\cdot(\pi x)'=-\frac{1}{2\sqrt{x}}-\frac{2}{\pi}\cdot \cos\pi x\cdot \pi =-\frac{1}{2\sqrt{x}}-2\cos\pi x](https://tex.z-dn.net/?f=f%27%28x%29%3D%283-%5Csqrt%7Bx%7D-%5Cfrac%7B2%7D%7B%5Cpi%7D%5Csin%20%5Cpi%20x%29%3D%283%29%27-%28%5Csqrt%7Bx%7D%29%27-%5Cfrac%7B2%7D%7B%5Cpi%7D%5Ccdot%28%5Csin%5Cpi%20x%29%27%3D%5C%5C%20%5C%5C%20%3D-%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D-%5Cfrac%7B2%7D%7B%5Cpi%7D%5Ccdot%20%5Ccos%5Cpi%20x%5Ccdot%28%5Cpi%20x%29%27%3D-%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D-%5Cfrac%7B2%7D%7B%5Cpi%7D%5Ccdot%20%5Ccos%5Cpi%20x%5Ccdot%20%5Cpi%20%3D-%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D-2%5Ccos%5Cpi%20x)
Значение производной функции в точке ![x_0=1](https://tex.z-dn.net/?f=x_0%3D1)
![f'(1)=-\frac{1}{2\cdot \sqrt{1}}-2\cos\pi=-0.5-2\cdot(-1)=-0.5+2=1.5](https://tex.z-dn.net/?f=f%27%281%29%3D-%5Cfrac%7B1%7D%7B2%5Ccdot%20%5Csqrt%7B1%7D%7D-2%5Ccos%5Cpi%3D-0.5-2%5Ccdot%28-1%29%3D-0.5%2B2%3D1.5)
Уравнение касательной:
![y=1.5(x-1)+2=1.5x-1.5+2=\boxed{1.5x+0.5}](https://tex.z-dn.net/?f=y%3D1.5%28x-1%29%2B2%3D1.5x-1.5%2B2%3D%5Cboxed%7B1.5x%2B0.5%7D)