Suppose that 2 ≤ f '(x) ≤ 4 for all values of x. What are the minimum and maximum possible values of f(6) − f(1)?

Question

Suppose that 2 ≤ f '(x) ≤ 4 for all values of x. What are the minimum and maximum possible values of f(6) − f(1)?

____≤ f(6) − f(1) ≤_______

Answer 1

By the Mean Value Theorem, there is at least one number, c, in the interval (1,6) such that

f'(c) = [f(6) - f(1)]/ (6 - 1)

So, f(6) - f(1) = 5f'(c).

Since 2 ≤ f'(c) ≤ 4, 10 ≤ 5f'(c) ≤ 20

So, f(6) - f(1) is between 10 and 20.