Уравнение касательной: y = f(x0) + f '(x0)(x – x0)).
f(x0) = 3sin(-Pi/2) + 12*(-Pi/2) = -3 - 6Pi
f'(x) = 3cosx + 12
f'(x0) = 3cos(-Pi/2) + 12 = 12
Подставляем полученные данные в уравнение касательной:
y = -3 - 6Pi + 12*(x + Pi/2) = -3 - 6Pi + 12x + 6Pi = 12x - 3 - уравнение касательной
1) (x-9)(2x+7)=0
x-9=0; 2x+7=0
x=9; 2x=-7
x=9; x=-3.5
Ответ: х = 9; - 3.5
2) -17х+3=-3х+2
-17х+3х=2-3
-14х=-1
х=1/14
Ответ: х=1/14
![\sin x \cos x( \sin^2x- \cos^2x)=\frac{\sqrt{3}}{8}\\ \frac{1}{2} \sin 2x \cdot (-\cos2x)=\frac{\sqrt{3}}{8}\\ \sin 4x=-\frac{\sqrt{3}}{2}\\ \left[\begin{array}{l} 4x=-\frac{2\pi }{3}+2\pi k ,\ k\in Z \\ 4x=-\frac{\pi }{3}+2\pi n ,\ n\in Z \end{array}\right\\ \left[\begin{array}{l} x=-\frac{\pi }{6}+\frac{\pi k}{2} ,\ k\in Z \\ x=-\frac{\pi }{12}+\frac{\pi n}{2},\ n\in Z \end{array}\right\\](https://tex.z-dn.net/?f=%5Csin+x+%5Ccos+x%28+%5Csin%5E2x-+%5Ccos%5E2x%29%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B8%7D%5C%5C+%5Cfrac%7B1%7D%7B2%7D+%5Csin+2x+%5Ccdot+%28-%5Ccos2x%29%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B8%7D%5C%5C+%5Csin+4x%3D-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5C%5C+%5Cleft%5B%5Cbegin%7Barray%7D%7Bl%7D+4x%3D-%5Cfrac%7B2%5Cpi+%7D%7B3%7D%2B2%5Cpi+k+%2C%5C+k%5Cin+Z+%5C%5C+4x%3D-%5Cfrac%7B%5Cpi+%7D%7B3%7D%2B2%5Cpi+n+%2C%5C+n%5Cin+Z+%5Cend%7Barray%7D%5Cright%5C%5C+%5Cleft%5B%5Cbegin%7Barray%7D%7Bl%7D+x%3D-%5Cfrac%7B%5Cpi+%7D%7B6%7D%2B%5Cfrac%7B%5Cpi+k%7D%7B2%7D+%2C%5C+k%5Cin+Z+%5C%5C+x%3D-%5Cfrac%7B%5Cpi+%7D%7B12%7D%2B%5Cfrac%7B%5Cpi+n%7D%7B2%7D%2C%5C+n%5Cin+Z+%5Cend%7Barray%7D%5Cright%5C%5C)
Найдем первые положительные корни в каждой серии решений:
![1)\ -\frac{\pi }{6}+\frac{\pi k}{2}>0,\ k\in Z\ \Rightarrow k>\frac{2}{3}, k\in Z\ \Rightarrow k=1\\ x=-\frac{\pi }{6}+\frac{\pi}{2}=\frac{\pi}{3}\\\\2)\ -\frac{\pi }{12}+\frac{\pi n}{2}>0,\ n\in Z\ \Rightarrow n>\frac{1}{6}, n\in Z\ \Rightarrow n=1\\ x=-\frac{\pi }{12}+\frac{\pi}{2}=\frac{5\pi}{12}](https://tex.z-dn.net/?f=1%29%5C+-%5Cfrac%7B%5Cpi+%7D%7B6%7D%2B%5Cfrac%7B%5Cpi+k%7D%7B2%7D%3E0%2C%5C+k%5Cin+Z%5C+%5CRightarrow+k%3E%5Cfrac%7B2%7D%7B3%7D%2C+k%5Cin+Z%5C+%5CRightarrow+k%3D1%5C%5C+x%3D-%5Cfrac%7B%5Cpi+%7D%7B6%7D%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cfrac%7B%5Cpi%7D%7B3%7D%5C%5C%5C%5C2%29%5C+-%5Cfrac%7B%5Cpi+%7D%7B12%7D%2B%5Cfrac%7B%5Cpi+n%7D%7B2%7D%3E0%2C%5C+n%5Cin+Z%5C+%5CRightarrow+n%3E%5Cfrac%7B1%7D%7B6%7D%2C+n%5Cin+Z%5C+%5CRightarrow+n%3D1%5C%5C+x%3D-%5Cfrac%7B%5Cpi+%7D%7B12%7D%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cfrac%7B5%5Cpi%7D%7B12%7D)
Сравним полученные положительные корни
![\frac{5\pi}{12} > \frac{\pi}{3}=\frac{4\pi}{12}](https://tex.z-dn.net/?f=%5Cfrac%7B5%5Cpi%7D%7B12%7D+%3E+%5Cfrac%7B%5Cpi%7D%7B3%7D%3D%5Cfrac%7B4%5Cpi%7D%7B12%7D)
Итак,
- наименьший положительный корень.
Ответ:
.
{3y+x=5
{13y-2x=11
{x+3y=5|*2
{-2x+13y=11
+{2x+6y=10
+{-2x+13y=11
19y=21:19
y=21/19
21/19*3+x=5
x=5-63/19
x=5-3 6/19
x=1 13/19
{7x+5y=17|*3
{8x-3y=5|*5
+{21x+15y=51
+{40x-15y=25
61x=76:61
x=76/61
76/61*7+5y=17
5y=17-553/61
5y=17-9 4/61
5y=7 57/61:5
y=96,8