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A1.
a)
б)
в)
г)
A2.
a)
б)![(frac{x^4-x^2}{x-1})'=frac{(x^4-x^2)'(x-1)-(x^4-x^2)(x-1)'}{(x-1)^2}=\
=frac{(4x^3-2x)(x-1)-(x^4-x^2)}{(x-1)^2}=frac{2x(2x^2-1)(x-1)-x^2(x-1)(x+1)}{(x-1)^2}=\
=frac{(x-1)(4x^3-2x-x^3-x^2)}{(x-1)^2}=frac{3x^3-x^2-2x}{x-1}=frac{x(3x^2-x-2)}{x-1}=frac{3x(x-1)(x+frac{2}{3})}{x-1}=\
=x(3x+2)](https://tex.z-dn.net/?f=%28frac%7Bx%5E4-x%5E2%7D%7Bx-1%7D%29%27%3Dfrac%7B%28x%5E4-x%5E2%29%27%28x-1%29-%28x%5E4-x%5E2%29%28x-1%29%27%7D%7B%28x-1%29%5E2%7D%3D%5C%0A%3Dfrac%7B%284x%5E3-2x%29%28x-1%29-%28x%5E4-x%5E2%29%7D%7B%28x-1%29%5E2%7D%3Dfrac%7B2x%282x%5E2-1%29%28x-1%29-x%5E2%28x-1%29%28x%2B1%29%7D%7B%28x-1%29%5E2%7D%3D%5C%0A%3Dfrac%7B%28x-1%29%284x%5E3-2x-x%5E3-x%5E2%29%7D%7B%28x-1%29%5E2%7D%3Dfrac%7B3x%5E3-x%5E2-2x%7D%7Bx-1%7D%3Dfrac%7Bx%283x%5E2-x-2%29%7D%7Bx-1%7D%3Dfrac%7B3x%28x-1%29%28x%2Bfrac%7B2%7D%7B3%7D%29%7D%7Bx-1%7D%3D%5C%0A%3Dx%283x%2B2%29 "(frac{x^4-x^2}{x-1})'=frac{(x^4-x^2)'(x-1)-(x^4-x^2)(x-1)'}{(x-1)^2}=\
=frac{(4x^3-2x)(x-1)-(x^4-x^2)}{(x-1)^2}=frac{2x(2x^2-1)(x-1)-x^2(x-1)(x+1)}{(x-1)^2}=\
=frac{(x-1)(4x^3-2x-x^3-x^2)}{(x-1)^2}=frac{3x^3-x^2-2x}{x-1}=frac{x(3x^2-x-2)}{x-1}=frac{3x(x-1)(x+frac{2}{3})}{x-1}=\
=x(3x+2)")
![f(x)=0.5cos(3x-frac{pi}{6}), x_0=frac{pi}{9}\
f'(x)=-0.5*3sin(3x-frac{pi}{6})=-1,5sin(3x-frac{pi}{6})\
f'(x_0)=-1.5*sin(frac{pi}{3}-frac{pi}{6})=-1.5sinfrac{pi}{6}=-1.5*0.5=-0.75](https://tex.z-dn.net/?f=f%28x%29%3D0.5cos%283x-frac%7Bpi%7D%7B6%7D%29%2C+x_0%3Dfrac%7Bpi%7D%7B9%7D%5C%0Af%27%28x%29%3D-0.5%2A3sin%283x-frac%7Bpi%7D%7B6%7D%29%3D-1%2C5sin%283x-frac%7Bpi%7D%7B6%7D%29%5C%0Af%27%28x_0%29%3D-1.5%2Asin%28frac%7Bpi%7D%7B3%7D-frac%7Bpi%7D%7B6%7D%29%3D-1.5sinfrac%7Bpi%7D%7B6%7D%3D-1.5%2A0.5%3D-0.75 "f(x)=0.5cos(3x-frac{pi}{6}), x_0=frac{pi}{9}\
f'(x)=-0.5*3sin(3x-frac{pi}{6})=-1,5sin(3x-frac{pi}{6})\
f'(x_0)=-1.5*sin(frac{pi}{3}-frac{pi}{6})=-1.5sinfrac{pi}{6}=-1.5*0.5=-0.75")
В1.
![f(x)=6x+xsqrt{x}=6x+x^{3/2}\
f'(x)=6+frac{3}{2}x^{1/2} textgreater 0|*2\
12+3sqrt{x} textgreater 0|:3\
4+sqrt{x} textgreater 0\
sqrt{x} textgreater -4\
xgeq 0](https://tex.z-dn.net/?f=f%28x%29%3D6x%2Bxsqrt%7Bx%7D%3D6x%2Bx%5E%7B3%2F2%7D%5C%0Af%27%28x%29%3D6%2Bfrac%7B3%7D%7B2%7Dx%5E%7B1%2F2%7D+textgreater++0%7C%2A2%5C%0A12%2B3sqrt%7Bx%7D+textgreater++0%7C%3A3%5C%0A4%2Bsqrt%7Bx%7D+textgreater++0%5C%0Asqrt%7Bx%7D+textgreater++-4%5C%0Axgeq+0%0A "f(x)=6x+xsqrt{x}=6x+x^{3/2}\
f'(x)=6+frac{3}{2}x^{1/2} textgreater 0|*2\
12+3sqrt{x} textgreater 0|:3\
4+sqrt{x} textgreater 0\
sqrt{x} textgreater -4\
xgeq 0")
В2.
![y=frac{x+4}{sqrt{x}}\
y'=frac{sqrt{x}-(x+4)frac{1}{2sqrt{x}}}{x}=frac{2x-x-4}{2xsqrt{x}}=frac{x-4}{2xsqrt{x}}](https://tex.z-dn.net/?f=y%3Dfrac%7Bx%2B4%7D%7Bsqrt%7Bx%7D%7D%5C%0Ay%27%3Dfrac%7Bsqrt%7Bx%7D-%28x%2B4%29frac%7B1%7D%7B2sqrt%7Bx%7D%7D%7D%7Bx%7D%3Dfrac%7B2x-x-4%7D%7B2xsqrt%7Bx%7D%7D%3Dfrac%7Bx-4%7D%7B2xsqrt%7Bx%7D%7D "y=frac{x+4}{sqrt{x}}\
y'=frac{sqrt{x}-(x+4)frac{1}{2sqrt{x}}}{x}=frac{2x-x-4}{2xsqrt{x}}=frac{x-4}{2xsqrt{x}}")
С1.
![y=(3-x)^4(2x+1)^3\
y'=4(3-x)^3(-1)(2x+1)^3+(3-x)^43(2x+1)^22=\=
-4(3-x)^3(2x+1)^3+6(3-x)^4(2x+1)^2=2(3-x)^3(2x+1)^2*\*(-2(2x+1)+3(3-x))=2(3-x)^3(2x+1)^2(-4x-2+9-3x)=\=
-2(x-3)^3(2x+1)^2(-7x+7)=14(x-3)^3(2x+1)^2(x-1)\
y' textless 0\
(x-3)^3(2x+1)^2(x-1) textless 0](https://tex.z-dn.net/?f=y%3D%283-x%29%5E4%282x%2B1%29%5E3%5C%0Ay%27%3D4%283-x%29%5E3%28-1%29%282x%2B1%29%5E3%2B%283-x%29%5E43%282x%2B1%29%5E22%3D%5C%3D%0A-4%283-x%29%5E3%282x%2B1%29%5E3%2B6%283-x%29%5E4%282x%2B1%29%5E2%3D2%283-x%29%5E3%282x%2B1%29%5E2%2A%5C%2A%28-2%282x%2B1%29%2B3%283-x%29%29%3D2%283-x%29%5E3%282x%2B1%29%5E2%28-4x-2%2B9-3x%29%3D%5C%3D%0A-2%28x-3%29%5E3%282x%2B1%29%5E2%28-7x%2B7%29%3D14%28x-3%29%5E3%282x%2B1%29%5E2%28x-1%29%5C%0Ay%27+textless++0%5C%0A%28x-3%29%5E3%282x%2B1%29%5E2%28x-1%29+textless++0 "y=(3-x)^4(2x+1)^3\
y'=4(3-x)^3(-1)(2x+1)^3+(3-x)^43(2x+1)^22=\=
-4(3-x)^3(2x+1)^3+6(3-x)^4(2x+1)^2=2(3-x)^3(2x+1)^2*\*(-2(2x+1)+3(3-x))=2(3-x)^3(2x+1)^2(-4x-2+9-3x)=\=
-2(x-3)^3(2x+1)^2(-7x+7)=14(x-3)^3(2x+1)^2(x-1)\
y' textless 0\
(x-3)^3(2x+1)^2(x-1) textless 0")
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A2.
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С1.
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