Find a polynomial P(x)=x^3+ax^2+bx+c satisfying ALL of the following properties:

local-maximum

#1

Find a polynomial P(x)=x^3+ax^2+bx+c satisfying ALL of the following properties:
i) x = -3 is a local maximum of P(x).
ii) x = 6 is a local minimum of P(x).
iii) P(0) = 0

Answer:

The derivative must be of the form:

dy/dx = (x+3)(x-6) = x^2 - 3x - 18

Integrate to get the function:

P(x)= (1/3)x^3 - (3/2)x^2 - 18x

And P(0) is 0.