a)dy/dx must have no solutions
dy/dx = ax^2+bx+c
Discriminant b^2 - 4ac < 0 for dy/dx means it has no solutions, so there are no stationary points on the cubic.
When a=1, b=2, c=2:
b^2-4ac = 4-8=-4
-4<0
y = x^2+2x+2 has no solutions
Integrate f(x) = x^2+2x+2 to get f(x) = 1/3x^3+x^2+2x+c
c can be 0, so f(x) = 1/3x^3+x^2+2x has no stationary points
b) dy/dx = 0 when x=2 and x=5
dy/dx = (x-2)(x-5) = x^2-7x+10
Integrate f(x) = x^2-7x+10 to get f(x) = 1/3x^3 - 7x^2 + 10x + c
c can be 0, so f(x) = 1/3x^3 - 7x^2 + 10x
c) f(x) is decreasing before x = -1, and increasing after x = -1.
dy/dx = x + 1, when x = -1 dy/dx = 0
Check to see if it's decreasing before and increasing after:
when x = -2, dy/dx < 0
when x = 0, dy/dx > 0
Local maximum at x = 5 (for example)
dy/dx = x-5:
x = 4, dy/dx = -1 which is < 0...this isn't right, f(x) should be increasing at x = 4.
Try dy/dx = -x+5, check for increasing/decreasing function:
when x = 4, dy/dx > 0
when x = 6, dy/dx < 0
dy/dx = (x+1)(-x+5)
= -x^2 + 4x + 5
Integrate f(x) = -x^2 + 4x + 5:
f(x) = -1/3x^3 + 2x^2 + 5x + c
c can be 0, so f(x) = -1/3x^3 + 2x^2 + 5x
d) f(x) is increasing before x = 4 and decreasing after x = 4, opposite for x = -2.
dy/dx = 0 when x = -2 and x = 4
dy/dx = x + 2
dy/dx = -x + 4 (checking either side of x, these equations match the increasing/decreasing functions)
dy/dx = (x+2)(-x+4)
= -x^2+2x+8
Integrate f(x) = -x^2+2x+8 to get f(x) = -1/3x^3 + x^2 + 8x + c
c can be 0, so f(x) = -1/3x^3 + x^2 + 8x