Calculus Help (deciphering a function and MVT)

This is the problem:

Let f be a function defined by f(x)= x3+ax2+bx+c and having the following properties:

I) The graph of f has a point of inflection at (0, -2).

II) The average (mean) value of f(x) on the closed interval [0,2] is -3

a) What are the values of a,b, and c?

b) Determine the value of x that satisfies the conclusion of the Mean Value Theorem for f on the closed interval [0,3]

I got that a=0 by calculating the second derivative, and then plugging in the inflection point. But I would really appreciate some help with the rest of the problem. Thanks to anyone who can help me out!!