(√32-3)²=(√32)²-2*3*√32+3²=32-6*√(16*2)+9=41-6*4√2=41-24√2
ОДЗ: х∈[-1;1]
![arccosx- \pi =arcsin \frac{4x}{3} \\ -( \pi -arccosx)=arcsin \frac{4x}{3} \\ \\ -arccos(-x)=arcsin \frac{4x}{3} \\ \\ sin(-arccos(-x))=sin(arcsin \frac{4x}{3} ) \\ \\-sin(arccos(-x))= \frac{4x}{3} \\ \\ - \sqrt{1-cos^2(arccos(-x))}=\frac{4x}{3} \\ \\ - \sqrt{1-(-x)^2} = \frac{4x}{3} \\ \\ - \sqrt{1- x^{2} } =\frac{4x}{3}](https://tex.z-dn.net/?f=arccosx-+%5Cpi+%3Darcsin+%5Cfrac%7B4x%7D%7B3%7D+%5C%5C+-%28+%5Cpi+-arccosx%29%3Darcsin+%5Cfrac%7B4x%7D%7B3%7D+%5C%5C+%5C%5C+-arccos%28-x%29%3Darcsin+%5Cfrac%7B4x%7D%7B3%7D+%5C%5C+%5C%5C+sin%28-arccos%28-x%29%29%3Dsin%28arcsin+%5Cfrac%7B4x%7D%7B3%7D+%29+%5C%5C+%5C%5C-sin%28arccos%28-x%29%29%3D+%5Cfrac%7B4x%7D%7B3%7D+++%5C%5C++%5C%5C+-+%5Csqrt%7B1-cos%5E2%28arccos%28-x%29%29%7D%3D%5Cfrac%7B4x%7D%7B3%7D+%5C%5C+%5C%5C++-+%5Csqrt%7B1-%28-x%29%5E2%7D+%3D+%5Cfrac%7B4x%7D%7B3%7D++%5C%5C++%5C%5C+-+%5Csqrt%7B1-+x%5E%7B2%7D+%7D+%3D%5Cfrac%7B4x%7D%7B3%7D)
![\sqrt{1-x^2} =- \frac{4x}{3} \ \ \ \textless \ =\ \textgreater \ \ \left \{ {{- \frac{4x}{3} \geq 0\ \ |*(-3)} \atop {1-x^2= \frac{16x^2}{9}|*9 }} \right. \\ \\ \left \{ {{4x \leq 0} \atop {9-9x^2=16 x^{2} }} \right. \ \ \ \textless \ =\ \textgreater \ \left \{ {{x \leq 0} \atop {16 x^{2} +9x^2-9=0}} \right. \\ \\ 16 x^{2} +9x^2-9=0 \\ \\25x^2=9\\ \\x^2= \frac{9}{25} \\ \\ x=^+_- \frac{3}{5} =^+_- 0.6](https://tex.z-dn.net/?f=%5Csqrt%7B1-x%5E2%7D+%3D-+%5Cfrac%7B4x%7D%7B3%7D+%5C+%5C++%5C+%5Ctextless+%5C+%3D%5C+%5Ctextgreater+%5C+%5C++%5Cleft+%5C%7B+%7B%7B-+%5Cfrac%7B4x%7D%7B3%7D+%5Cgeq+0%5C+%5C++%7C%2A%28-3%29%7D+%5Catop+%7B1-x%5E2%3D+%5Cfrac%7B16x%5E2%7D%7B9%7D%7C%2A9+%7D%7D+%5Cright.++%5C%5C++%5C%5C++%5Cleft+%5C%7B+%7B%7B4x+%5Cleq+0%7D+%5Catop+%7B9-9x%5E2%3D16+x%5E%7B2%7D+%7D%7D+%5Cright.+%5C+%5C+%5C+%5Ctextless+%5C+%3D%5C+%5Ctextgreater+%5C++%5Cleft+%5C%7B+%7B%7Bx+%5Cleq+0%7D+%5Catop+%7B16+x%5E%7B2%7D+%2B9x%5E2-9%3D0%7D%7D+%5Cright.++%5C%5C++%5C%5C+16+x%5E%7B2%7D+%2B9x%5E2-9%3D0+%5C%5C++%5C%5C25x%5E2%3D9%5C%5C+%5C%5Cx%5E2%3D+%5Cfrac%7B9%7D%7B25%7D++%5C%5C++%5C%5C+x%3D%5E%2B_-+%5Cfrac%7B3%7D%7B5%7D+%3D%5E%2B_-+0.6)
С учетом ОДЗ и с учетом системы x≤0, подходит только корень -0,6
ОТВЕТ: -0,6
![y=x^{4}-2x^{3}+3](https://tex.z-dn.net/?f=y%3Dx%5E%7B4%7D-2x%5E%7B3%7D%2B3)
Составим уравнение касательной в точке х=1/2
![y`=4x^{3}-6x^{2}=2x^{2}(2x-3)](https://tex.z-dn.net/?f=y%60%3D4x%5E%7B3%7D-6x%5E%7B2%7D%3D2x%5E%7B2%7D%282x-3%29)
![y(\frac{1}{2})=(\frac{1}{2})^{4}-2(\frac{1}{2})^{3}+3=\frac{1}{16}-\frac{2}{8}+3=\frac{1-4+48}{16}=\frac{45}{16}](https://tex.z-dn.net/?f=y%28%5Cfrac%7B1%7D%7B2%7D%29%3D%28%5Cfrac%7B1%7D%7B2%7D%29%5E%7B4%7D-2%28%5Cfrac%7B1%7D%7B2%7D%29%5E%7B3%7D%2B3%3D%5Cfrac%7B1%7D%7B16%7D-%5Cfrac%7B2%7D%7B8%7D%2B3%3D%5Cfrac%7B1-4%2B48%7D%7B16%7D%3D%5Cfrac%7B45%7D%7B16%7D)
![y`(\frac{1}{2})=2(\frac{1}{2})^{2}(2*\frac{1}{2}-3)=\frac{1}{2}(1-3)=\frac{1}{2}*(-2)=-1](https://tex.z-dn.net/?f=y%60%28%5Cfrac%7B1%7D%7B2%7D%29%3D2%28%5Cfrac%7B1%7D%7B2%7D%29%5E%7B2%7D%282%2A%5Cfrac%7B1%7D%7B2%7D-3%29%3D%5Cfrac%7B1%7D%7B2%7D%281-3%29%3D%5Cfrac%7B1%7D%7B2%7D%2A%28-2%29%3D-1)
Уравнение касательной в общем виде:
![y=y(x_{0})+y`(x_{0})(x-x_{0})](https://tex.z-dn.net/?f=y%3Dy%28x_%7B0%7D%29%2By%60%28x_%7B0%7D%29%28x-x_%7B0%7D%29)
Подставим найденные значения в это уравнение:
![y=\frac{45}{16}+(-1)(x-\frac{1}{2})=\frac{45}{16}-x+\frac{1}{2}=\frac{53}{16}-x=-x+3\frac{5}{16}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B45%7D%7B16%7D%2B%28-1%29%28x-%5Cfrac%7B1%7D%7B2%7D%29%3D%5Cfrac%7B45%7D%7B16%7D-x%2B%5Cfrac%7B1%7D%7B2%7D%3D%5Cfrac%7B53%7D%7B16%7D-x%3D-x%2B3%5Cfrac%7B5%7D%7B16%7D)
- уравнение касательной
Найдём угол между этой касательной и осью ОХ.
k=-1 (Коэффициент при х в уравнении касательной)
tga=k
tga=-1
a=135 градусов
F`(x)=[(x+1)`(x²+2x-1)-(x+1)*(x²+2x-1)`]/(x²+2x-1)²=
=(x²+2x-1-2x²-4x-2)(x²+2x-1)²=(-x²-2x-3)(x²+2x-1)²
f`(-2,5)=(-6,25+5-3)/(6,25-5-1)²=-4,25/0,0625=-68