Calculus I question

Question

If f(2) = 4 and f '(x) ≥ 2 for 2 ≤ x ≤ 7, how small can f(7) possibly be?

Answer 1

By the Mean value Theorem, there exists a number c such that f(7)-f(2)=f'(c)(7-2). So f(7)=4+5f'(c).

We are given that f'(x)>=2, so f(7)>= 4+2*5=14. The smallest possible value for f(7) is 14.