calculus 1- Mean Value Theorem

Question

In each part of this problem a value for a function f(x) at a is given, a range for the derivative of f (x) on (a, b) is given, and a possible value for f (b) is given. Assuming that

f (a) is correct and the range for f ′ (x) is correct, can f (b) be correct?

f(1) = 2, f′(x) ∈ [−1,1], f(5) = 0

Answer 1

By the MVT, there exists at least one number c in the interval (1,5) so that f'(c) = [f(5) - f(1)] / (5-1)

f'(c) = [0 - 2] / (5 - 1) = -1/2, which is in the interval [-1,1].

So, f(5) could be as given.