1) y'=(-x^3+0,5x^2-x+1)'= -3x^2+x-1
2) y'=(-3cosx)' * (x^2+2) + (-3cosx)*(x^2+2)'=3sinx(x^2+2)-3cosx*2x=3sinx(x^2+2)-6xcosx
3) y'= (x^-1/2)'=-1/2 * x^-3/2 = -1/(2x^(3/2))
4) y'=(1/sinx)'=(1'*sinx - 1*sinx')/sin^2(x) = (0-cosx)/sin^2(x) = -cosx/sin^2(x) = -ctgx/sinx
5) y'=x^4/(3-x)=(x^4'(3-x)-x^4(3-x)')/(3-x)^2=(4x^3(3-x)+x^4)/(3-x)^2=(12x^3-4x^4+x^4)/(3-x)^2= (12x^3-3x^4)/(3-x)^2
6) y'= (x^2+ctgx)' = 2x - 1/sin^2(x)
1) y' = -2x^3 + x^2 -2
2) y'=(4sqrt(x)+3)'(1-1/x)+(4sqrt(x)+3)(1-1/x)'=(2/sqrt(x))(1-1/x)+(4sqrt(x)+3)(1/x^2)
3) y'=(-x^-3)'=3x^-4=3/x^4
4) y'=(3'*sinx - 3sinx')/sin^2(x)=(0-3cosx)/sin^2(x))=-3cosx/sin^2(x)=-3ctgx/sinx
5) y'=((x^2+4)'cosx - (x^2+4)cosx')/cos^2(x)=(2xcosx + sinx(x^2+4)/cos^2(x)
6) y'=x^2' *tgx + x^2 * tgx' = 2xtgx + x^2/cos^2(x)
1. находим производную:
![f'(x)=2x-8-{10\over2\sqrt{x^2-8x+12}}\cdot(2x-8)\\ f'(x)=2x-8-{5\cdot(2x-8)\over\sqrt{x^2-8x+12}}\\ f'(x)={(2x-8)\cdot(\sqrt{x^2-8x+12}-5)\over\sqrt{x^2-8x+12}}](https://tex.z-dn.net/?f=f%27%28x%29%3D2x-8-%7B10%5Cover2%5Csqrt%7Bx%5E2-8x%2B12%7D%7D%5Ccdot%282x-8%29%5C%5C+f%27%28x%29%3D2x-8-%7B5%5Ccdot%282x-8%29%5Cover%5Csqrt%7Bx%5E2-8x%2B12%7D%7D%5C%5C+f%27%28x%29%3D%7B%282x-8%29%5Ccdot%28%5Csqrt%7Bx%5E2-8x%2B12%7D-5%29%5Cover%5Csqrt%7Bx%5E2-8x%2B12%7D%7D)
2. находим критические точки (точки, в которых производная равна нулю или не существует):
![f'(x)={(2x-8)\cdot(\sqrt{x^2-8x+12}-5)\over\sqrt{x^2-8x+12}}\\ (2x-8)\cdot(\sqrt{x^2-8x+12}-5)=0\\ 2x-8=0\\ x=4\\ \sqrt{x^2-8x+12}-5=0\\ x^2-8x+12=25\\ x^2-8x-13=0\\ D=64+52=116\\ x_1=4-\sqrt{29},\; x_2=4+\sqrt{29}\\ \sqrt{x^2-8x+12}=0\\ x^2-8x+12=0\\ x_1=2,\; x_2=6\\](https://tex.z-dn.net/?f=f%27%28x%29%3D%7B%282x-8%29%5Ccdot%28%5Csqrt%7Bx%5E2-8x%2B12%7D-5%29%5Cover%5Csqrt%7Bx%5E2-8x%2B12%7D%7D%5C%5C+%282x-8%29%5Ccdot%28%5Csqrt%7Bx%5E2-8x%2B12%7D-5%29%3D0%5C%5C+2x-8%3D0%5C%5C+x%3D4%5C%5C+%5Csqrt%7Bx%5E2-8x%2B12%7D-5%3D0%5C%5C+x%5E2-8x%2B12%3D25%5C%5C+x%5E2-8x-13%3D0%5C%5C+D%3D64%2B52%3D116%5C%5C+x_1%3D4-%5Csqrt%7B29%7D%2C%5C%3B+x_2%3D4%2B%5Csqrt%7B29%7D%5C%5C+%5Csqrt%7Bx%5E2-8x%2B12%7D%3D0%5C%5C+x%5E2-8x%2B12%3D0%5C%5C+x_1%3D2%2C%5C%3B+x_2%3D6%5C%5C)
3. Отмечаем полученные точки на числовой прямой и смотрим знаки производной на промежутках:
---------_________+++++___разрыв___разрыв___---------_________++++++
↓ 4-sqrt(29) ↑ 2 4 6 ↓ 4+sqrt(29) ↑
xmin1=4-sqrt(29)
xmin2=4+sqrt(29)
y(min1)=y(4-sqrt(29))= -25
y(min2)=y(4+sqrt(29))= -25
Ответ: наименьшее значение функции равно -25