2Х + 3Y = - 12
2X = - 12 - 3Y
4X = - 24 - 6Y
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4X - 6Y = 0
4X = 6Y
------------
4X = 4X
- 24 - 6Y = 6Y
12Y = - 24
Y = - 2
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4X = 6Y
4X = - 12
X = - 3
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Ответ ( - 3 ; - 2 )
<u>В обоих случая разность косинусов.</u>
![cos( \alpha + \frac{ \pi }{6} )-cos( \alpha - \frac{ \pi }{6} )=-2sin \frac{2 \alpha }{2} *sin \frac{ \frac{ \pi }{3} }{2} = \\ \\ -2sin \alpha *sin \frac{ \pi }{6} =-2sin \alpha * \frac{1}{2} =-sin \alpha \\ \\ \\ cos( \frac{ \pi }{3} - \alpha )-cos( \frac{ \pi }{3} + \alpha )= \\ \\ -2sin \frac{ (\frac{ 2\pi }{3}) }{2} * sin( \frac{-2 \alpha }{2} ) =-2sin \frac{ \pi }{3} *sin(- \alpha )= \\ \\ -2* \frac{ \sqrt{3} }{2} *(-sin \alpha )= \sqrt{3} sin \alpha](https://tex.z-dn.net/?f=cos%28%20%5Calpha%20%2B%20%5Cfrac%7B%20%5Cpi%20%7D%7B6%7D%20%29-cos%28%20%5Calpha%20-%20%5Cfrac%7B%20%5Cpi%20%7D%7B6%7D%20%29%3D-2sin%20%5Cfrac%7B2%20%5Calpha%20%7D%7B2%7D%20%2Asin%20%5Cfrac%7B%20%5Cfrac%7B%20%5Cpi%20%7D%7B3%7D%20%7D%7B2%7D%20%3D%20%5C%5C%20%5C%5C%20-2sin%20%5Calpha%20%2Asin%20%5Cfrac%7B%20%5Cpi%20%7D%7B6%7D%20%3D-2sin%20%5Calpha%20%2A%20%5Cfrac%7B1%7D%7B2%7D%20%3D-sin%20%5Calpha%20%5C%5C%20%5C%5C%20%5C%5C%20cos%28%20%5Cfrac%7B%20%5Cpi%20%7D%7B3%7D%20-%20%5Calpha%20%29-cos%28%20%5Cfrac%7B%20%5Cpi%20%7D%7B3%7D%20%2B%20%5Calpha%20%29%3D%20%5C%5C%20%5C%5C%20-2sin%20%5Cfrac%7B%20%28%5Cfrac%7B%202%5Cpi%20%7D%7B3%7D%29%20%7D%7B2%7D%20%2A%20sin%28%20%5Cfrac%7B-2%20%5Calpha%20%7D%7B2%7D%20%29%20%3D-2sin%20%5Cfrac%7B%20%5Cpi%20%7D%7B3%7D%20%2Asin%28-%20%5Calpha%20%29%3D%20%5C%5C%20%5C%5C%20-2%2A%20%5Cfrac%7B%20%5Csqrt%7B3%7D%20%7D%7B2%7D%20%2A%28-sin%20%5Calpha%20%29%3D%20%5Csqrt%7B3%7D%20sin%20%5Calpha)
![1)f(x)=3x-\frac{1}{3}x^{3}-x^{2}\\\\f'(x)=3(x)'-\frac{1}{3}(x^{3})'-(x^{2})'=3-\frac{1}{3}*3x^{2}-2x=3-x^{2}-2x\\\\f'(x)<0\Rightarrow -x^{2}-2x+3<0\\\\x^{2}+2x-3>0\\\\(x-1)(x+3)>0](https://tex.z-dn.net/?f=1%29f%28x%29%3D3x-%5Cfrac%7B1%7D%7B3%7Dx%5E%7B3%7D-x%5E%7B2%7D%5C%5C%5C%5Cf%27%28x%29%3D3%28x%29%27-%5Cfrac%7B1%7D%7B3%7D%28x%5E%7B3%7D%29%27-%28x%5E%7B2%7D%29%27%3D3-%5Cfrac%7B1%7D%7B3%7D%2A3x%5E%7B2%7D-2x%3D3-x%5E%7B2%7D-2x%5C%5C%5C%5Cf%27%28x%29%3C0%5CRightarrow%20-x%5E%7B2%7D-2x%2B3%3C0%5C%5C%5C%5Cx%5E%7B2%7D%2B2x-3%3E0%5C%5C%5C%5C%28x-1%29%28x%2B3%29%3E0)
+ - +
________₀_________₀_________
- 3 1
///////////////// ////////////////////
Ответ : x ∈ (- ∞ ; - 3) ∪ (1 ; + ∞)
![2)f(x)=\frac{2x}{(x-1)^{2} }\\\\f'(x)=\frac{2(x)'*(x-1)^{2}-2x*((x-1)^{2})'}{(x-1)^{4}}=\frac{2*(x^{2}-2x+1)-2x*2(x-1)}{(x-1)^{4}}=\frac{2x^{2}-4x+2-4x^{2}+4x}{(x-1)^{4}}=\frac{-2x^{2}+2 }{(x-1)^{4}}=\frac{-2(x^{2}-1)}{(x-1)x^{4}}\\\\f'(x)=0\Rightarrow\\\\\left \{ {{x^{2}-1=0 } \atop {x-1\neq0 }} \right. \\\\\\\left \{ {{(x-1)(x+1)=0} \atop {x\neq1 }} \right. \\\\\\\left \{ {{\left[\begin{array}{ccc}x_{1}=-1 \\x_{2} =1-neyd\end{array}\right } \atop {x\neq1 }} \right. \\\\](https://tex.z-dn.net/?f=2%29f%28x%29%3D%5Cfrac%7B2x%7D%7B%28x-1%29%5E%7B2%7D%20%7D%5C%5C%5C%5Cf%27%28x%29%3D%5Cfrac%7B2%28x%29%27%2A%28x-1%29%5E%7B2%7D-2x%2A%28%28x-1%29%5E%7B2%7D%29%27%7D%7B%28x-1%29%5E%7B4%7D%7D%3D%5Cfrac%7B2%2A%28x%5E%7B2%7D-2x%2B1%29-2x%2A2%28x-1%29%7D%7B%28x-1%29%5E%7B4%7D%7D%3D%5Cfrac%7B2x%5E%7B2%7D-4x%2B2-4x%5E%7B2%7D%2B4x%7D%7B%28x-1%29%5E%7B4%7D%7D%3D%5Cfrac%7B-2x%5E%7B2%7D%2B2%20%7D%7B%28x-1%29%5E%7B4%7D%7D%3D%5Cfrac%7B-2%28x%5E%7B2%7D-1%29%7D%7B%28x-1%29x%5E%7B4%7D%7D%5C%5C%5C%5Cf%27%28x%29%3D0%5CRightarrow%5C%5C%5C%5C%5Cleft%20%5C%7B%20%7B%7Bx%5E%7B2%7D-1%3D0%20%7D%20%5Catop%20%7Bx-1%5Cneq0%20%7D%7D%20%5Cright.%20%5C%5C%5C%5C%5C%5C%5Cleft%20%5C%7B%20%7B%7B%28x-1%29%28x%2B1%29%3D0%7D%20%5Catop%20%7Bx%5Cneq1%20%7D%7D%20%5Cright.%20%5C%5C%5C%5C%5C%5C%5Cleft%20%5C%7B%20%7B%7B%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx_%7B1%7D%3D-1%20%5C%5Cx_%7B2%7D%20%3D1-neyd%5Cend%7Barray%7D%5Cright%20%7D%20%5Catop%20%7Bx%5Cneq1%20%7D%7D%20%5Cright.%20%5C%5C%5C%5C)
![Otvet:\boxed{-1}](https://tex.z-dn.net/?f=Otvet%3A%5Cboxed%7B-1%7D)
![3)f(x)=x*Cos2x\\\\f'(x)=(x)'*Cos2x+x*(Cos2x)'=1*Cos2x+x*(-2Sin2x)=Cos2x-2xSin2x\\\\f'(0)=Cos0-2*0*Sin0=1-0=1](https://tex.z-dn.net/?f=3%29f%28x%29%3Dx%2ACos2x%5C%5C%5C%5Cf%27%28x%29%3D%28x%29%27%2ACos2x%2Bx%2A%28Cos2x%29%27%3D1%2ACos2x%2Bx%2A%28-2Sin2x%29%3DCos2x-2xSin2x%5C%5C%5C%5Cf%27%280%29%3DCos0-2%2A0%2ASin0%3D1-0%3D1)
3√5∙7√2∙√10.=21·\/5\/2·\/10=21·10=210.