4.
S=t⁵ - t³
V(t)=S ' =5t⁴ - 3t²
V(2)=5*2⁴ - 3*2² = 5*16 - 3*4 = 80-12= 68 (м/с)
5.
f(x)=12x-x³
f ' (x)=12 -3x²
12-3x²≤0
3x²-12≥0
3(x²-4)≥0
x²-4≥0
(x-2)(x+2)≥0
x=2 x= -2
+ - +
------ -2 ----------- 2 ------------
\\\\\\\\ \\\\\\\\\\\\\
x∈(-∞; -2]U[2; +∞)
6.
f(x)=cos2x+x√3
f ' (x)= -2sin2x +√3
-2sin2x+√3=0
-2sin2x= -√3
sin2x= √3/2
2x= (-1)ⁿ (π/3) + πn
x=(-1)ⁿ (π/6) + (π/2)n, n∈Z
При n=0 x=π/6
При n=1 x= -π/6 + π/2 = 2π/6 = π/3
При n=2 x= π/6 + π = 7π/6
При n=3 x= -π/6 + 3π/2 = 8π/6=4π/3
При n=4 x=π/6 + 2π= 13π/6
При n=5 x=-π/6 + 5π/2 = 14π/6=7π/3
При n=6 x=π/6 + 3π= 19π/6
При n=7 x=-π/6 +7π/2=20π/6=10π/3
Ответ: π/6; π/3; 7π/6; 4π/3; 13π/6; 7π/3; 19π/6; 10π/3.