Input
x^3 + 3 x^2 – 4 x – 12
Polynomial discriminant
Delta = 400
Derivative
d/dx(x^3 + 3 x^2 – 4 x – 12) = 3 x^2 + 6 x – 4
Indefinite integral
integral (-12 – 4 x + 3 x^2 + x^3) dx = x^4/4 + x^3 – 2 x^2 – 12 x + constant
Local maximum
max{x^3 + 3 x^2 – 4 x – 12} = (14 sqrt(7/3))/3 – 6 at x = -1 – sqrt(7/3)
Local minimum
min{x^3 + 3 x^2 – 4 x – 12} = -2/9 (27 + 7 sqrt(21)) at x = sqrt(7/3) – 1
Definite integral area below the axis between the smallest and largest real roots
integral_(-3)^2 (-12 – 4 x + 3 x^2 + x^3) theta(12 + 4 x – 3 x^2 – x^3) dx = -32
Definite integral area above the axis between the smallest and largest real roots
integral_(-3)^2 (-12 – 4 x + 3 x^2 + x^3) theta(-12 – 4 x + 3 x^2 + x^3) dx = 3/4 = 0.75