1.
ОДЗ: x≠0.4
(x²+10x)(2-5x)≤0
x(x+10)*(-5)(x-0.4)≤0
x(x+10)(x-0.4)≥0
x=0 x= -10 x=0.4
- + - +
--------- -10 ----------- 0 ------------ 0.4 -----------
\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\
x∈[-10; 0]U(0.4; +∞)
2.
a) cos²x+cosx= -sin²x
cos²x+sin²x+cosx=0
1+cosx=0
cosx= -1
x=π + 2πk, k∈Z.
б) cos2x=1+4cosx
cos²x-sin²x-4cosx-1=0
cos²x-(1-cos²x)-4cosx-1=0
2cos²x-1-4cosx-1=0
2cos²x-4cosx-2=0
cos²x-2cosx-1=0
y=cosx
y²-2y-1=0
D=4+4=8=(√8)²=(2√2)²
y₁=(2-2√2)/2=1-√2 cosx=1-√2 x=(+/-)arccos(1-√2) + 2πk, k∈Z
y₂=1+√2 cosx=1+√2 |1+√2|>1 нет решений.
3.
a) y ' = (2/cos²x) - 6x
б) y ' = - 3/(2√(4-3x))
в) y ' = 3x²sin2x + 2x³cos2x
4.
y=3+7x-4x²
y ' = -8x+7
k= -9
-8x+7= -9
-8x= -9-7
-8x= -16
x=2 - точка касания.
у(2) = 3+7*2 -4*2²=3+14-16=1
(2; 1) - координаты точки касания.