Question

find where the average rate of change is the same as the instantaneous rate of change for
f(x)=x^3+x=1 over [0,2]

Answer 1

Use LMVT(Lagrange Mean Value theorem)
f'(c)=((f(b)-f(a))/(b-a)
3c+1=(11-(-1))/(2-0)
3c+1=6
c=5/3

Answer 2

So that's all I have to do?

Answer 3

Yes