Calculus Question Please help Me Understand

Question

Suppose that 4 ≤ f '(x) ≤ 5 for all values of x. What are the minimum and maximum possible values of f(4) − f(1)?

Answer 1

By the Mean Value Theorem, there is at least one real number c in the interval (1,4) such that

f'(c) = [f(4) - f(1)] / (4-1)

So, f(4) - f(1) = 3f'(c)

But, since 4 ≤ f'(c) ≤ 5, we have 12 ≤ 3f'(c) ≤ 15

So, 12 ≤ f(4) - f(1) ≤ 15