![\frac{(x-2)^2+4x-5}{(x-5)^3} \geq 0](https://tex.z-dn.net/?f=+%5Cfrac%7B%28x-2%29%5E2%2B4x-5%7D%7B%28x-5%29%5E3%7D++%5Cgeq+0)
На промежутке [-3; 8]
![\frac{x^2-4x+4+4x-5}{(x-5)^3} \geq 0](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2-4x%2B4%2B4x-5%7D%7B%28x-5%29%5E3%7D+%5Cgeq+0)
![\frac{x^2-1}{(x-5)^3} \geq 0](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2-1%7D%7B%28x-5%29%5E3%7D+%5Cgeq+0)
![\frac{(x-1)(x+1)}{(x-5)^3} \geq 0](https://tex.z-dn.net/?f=+%5Cfrac%7B%28x-1%29%28x%2B1%29%7D%7B%28x-5%29%5E3%7D+%5Cgeq+0+)
По методу интервалов
x ∈ [-1; 1] U (5; +oo)
На промежутке [-3; 8] будет
x ∈ [-1; 1] U (5; 8]
Целые решения: -1, 0, 1, 6, 7, 8.
Всего 6 решений.
Используем формулы
синус двойного угла
![2 sin(x) cos (x)=sin(2x)](https://tex.z-dn.net/?f=2+sin%28x%29+cos+%28x%29%3Dsin%282x%29)
основное тригонометрическое тождество
![sin^2(x)+cos^2(x)=1](https://tex.z-dn.net/?f=sin%5E2%28x%29%2Bcos%5E2%28x%29%3D1)
квадрат двучлена
![(x+y)^2=x^2+2xy+y^2](https://tex.z-dn.net/?f=%28x%2By%29%5E2%3Dx%5E2%2B2xy%2By%5E2)
сумма кубов
![a^3+b^3=(a+b)(a^2-ab+b^2)](https://tex.z-dn.net/?f=a%5E3%2Bb%5E3%3D%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29)
![sin a+cos a=\frac{1}{2}](https://tex.z-dn.net/?f=sin+a%2Bcos+a%3D%5Cfrac%7B1%7D%7B2%7D)
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![(sin a+ cos a)^2=(\frac{1}{2})^2](https://tex.z-dn.net/?f=%28sin+a%2B+cos+a%29%5E2%3D%28%5Cfrac%7B1%7D%7B2%7D%29%5E2)
![sin^2 a+2sin a cos a+cos^2 a=\frac{1}{4}](https://tex.z-dn.net/?f=sin%5E2+a%2B2sin+a+cos+a%2Bcos%5E2+a%3D%5Cfrac%7B1%7D%7B4%7D)
![1+sin(2a)=\frac{1}{4}](https://tex.z-dn.net/?f=1%2Bsin%282a%29%3D%5Cfrac%7B1%7D%7B4%7D)
![sin(2a)=-\frac{3}{4}](https://tex.z-dn.net/?f=sin%282a%29%3D-%5Cfrac%7B3%7D%7B4%7D)
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![sin^3a+cos^3a=(sin a+cos a)(sin^2 a-sin a*cos a+cos^2 a)=\\\\(sin a+cos a)(1-\frac{1}{2}*sin(2a))=\\\\\frac{1}{2}(1-\frac{1}{2}*\frac{-3}{4})=\frac{1}{2}*(1+\frac{3}{8})=\frac{1}{2}*\frac{11}{8}=\frac{11}{16}](https://tex.z-dn.net/?f=sin%5E3a%2Bcos%5E3a%3D%28sin+a%2Bcos+a%29%28sin%5E2+a-sin+a%2Acos+a%2Bcos%5E2+a%29%3D%5C%5C%5C%5C%28sin+a%2Bcos+a%29%281-%5Cfrac%7B1%7D%7B2%7D%2Asin%282a%29%29%3D%5C%5C%5C%5C%5Cfrac%7B1%7D%7B2%7D%281-%5Cfrac%7B1%7D%7B2%7D%2A%5Cfrac%7B-3%7D%7B4%7D%29%3D%5Cfrac%7B1%7D%7B2%7D%2A%281%2B%5Cfrac%7B3%7D%7B8%7D%29%3D%5Cfrac%7B1%7D%7B2%7D%2A%5Cfrac%7B11%7D%7B8%7D%3D%5Cfrac%7B11%7D%7B16%7D)
Разложим выражение на сложение суммы и разницы двух одночленов:
(2x^5-5y^6)(2x^5+5y^6)=4x^10-25y^12.