1. D=1-(4*(-56)) = 225
X1= -(1+√(225))/2=-8
X2 =-(1-√(225))/2 = 7
или
2. По теореме Виета
x1+x2 = -b/a = -1/1= -1
x1*x2 = c/a = -56/1 = -56
X1=-8
X2 = 7
![y=3x^4+8x^3-18x^2+1 \\ \\ y'=12x^3+24x^2-36x=12x(x^2+2x-3) \\ \\ y'=0 \\ 12x(x^2+2x-3)=0 \\ x_1=0 \\ x^2+2x-3=0 \\ D=4+12=16 \\ x= \frac{-2+-4}{2} = \left \{ {{x_2=-3} \atop {x_3=1}} \right. \\ \\ y'=12x(x+3)(x-1)](https://tex.z-dn.net/?f=y%3D3x%5E4%2B8x%5E3-18x%5E2%2B1+%5C%5C++%5C%5C+y%27%3D12x%5E3%2B24x%5E2-36x%3D12x%28x%5E2%2B2x-3%29+%5C%5C++%5C%5C+y%27%3D0+%5C%5C+12x%28x%5E2%2B2x-3%29%3D0+%5C%5C+x_1%3D0+%5C%5C+x%5E2%2B2x-3%3D0+%5C%5C+D%3D4%2B12%3D16+%5C%5C+x%3D+%5Cfrac%7B-2%2B-4%7D%7B2%7D+%3D+%5Cleft+%5C%7B+%7B%7Bx_2%3D-3%7D+%5Catop+%7Bx_3%3D1%7D%7D+%5Cright.++%5C%5C++%5C%5C+y%27%3D12x%28x%2B3%29%28x-1%29)
- + - +
-----------------|-----------------|-----------------|---------------->x
-3 0 1
min max min
![y(-3)=3(-3)^4+8(-3)^3-18(-3)^2+1=243-216-162+1=-134 \\ \\ y(0)=3*0+8*0-18*0+1=1 \\ \\ y(1)=3*1+8*1-18*1+1=-6](https://tex.z-dn.net/?f=y%28-3%29%3D3%28-3%29%5E4%2B8%28-3%29%5E3-18%28-3%29%5E2%2B1%3D243-216-162%2B1%3D-134+%5C%5C++%5C%5C+y%280%29%3D3%2A0%2B8%2A0-18%2A0%2B1%3D1+%5C%5C++%5C%5C+y%281%29%3D3%2A1%2B8%2A1-18%2A1%2B1%3D-6)
функция убывает на промежутке x∈
![(-\infty; -3)U(0;1)](https://tex.z-dn.net/?f=%28-%5Cinfty%3B+-3%29U%280%3B1%29)
функция возрастает на промежутке x∈
![(-3;0)U(1;+\infty)](https://tex.z-dn.net/?f=%28-3%3B0%29U%281%3B%2B%5Cinfty%29)
точка максимума (0;1)
точки минимума (-3;-134) и (1;-6)
1)7a+14/а2-4=7(а+2)/(а-2)(а+2)=7/а-2
2)х-8/х2-16х+64=х-8/(х-8)2=х-8
3)3х2-2х-8=0 D=(-2)2-4*3*-8=4+96=100 х1=2+10/6=2 х2=2-10/6=-1целая1/3
(x^3+3^3)/(x-3) - x^2 - 3x -9 =(x+3)(x^2-3x+9)/(x-3) -(x^2+3x+9)=
<u>(x+3)(x^2-3x+9) -(x^2+3x+9)*(x-3)</u> =
x-3
<u>x^3-3x^2+9x+3x^2-9x+27 - x^3+3x^2-3x^2+9x-9x+27</u>=
x-3
<u>
27 +27</u>=
x-3
<u>
54 </u>
x-3
![y=x^2-8x+7](https://tex.z-dn.net/?f=y%3Dx%5E2-8x%2B7)
Это уравнение параболы. Т.к. старший коэффициент а =1 > 0, то ветви параболы направлены вверх, тогда в вершине функция будет иметь наименьшее значение.
Координаты вершины параболы
![x_0 = - \frac{b}{2a} = - \frac{-8}{2*1} = 4](https://tex.z-dn.net/?f=x_0+%3D+-++%5Cfrac%7Bb%7D%7B2a%7D+%3D+-++%5Cfrac%7B-8%7D%7B2%2A1%7D+%3D+4)
<span>Найдем наименьшие значение функции
![y(4)=4^2-8*4+7 = -9](https://tex.z-dn.net/?f=y%284%29%3D4%5E2-8%2A4%2B7+%3D+-9)
Ответ: -9</span><span />