I know that the mean value theorem applies when an interval [a,b] is continous and (a,b) is differentiable. When it does, there is a f(c) where f'(c) = f(b) -f(a) divided by (b-a)

I have the pic of the problem below for problem 1a and 1b

How am i suppose to find minimum possible value of f(2) if f is differentiable with f(0) = 1 and f'(x) >= -4 for -9 <= <=9?

I'm not sure what to do since the slope if variable. is it possible to calculate f(x) when slope is -4? I'm not sure what to do and dont understand what they are asking.