calculus question?

Question

verify f(x)= x/x+1 satisfies the hypothesis of the mean value theorem on the interval [0,2]. Then find all values of c that satisfy the conclusion of the theorem.

please show as much work as possible!!

Answer 1

f(x) = x/x+1 = 1 + 1 = 2
so f satisfies the hypothesis

Answer 2

Clearly f is continuous on [0,2] and differentiable on (0,2). The Mean Value Theorem says there is a number c in (0,2) such that f(c) = [f(2) - f(0)]/(2 - 0). Thus, we are looking for those c such that c/(c + 1) = (2/3 - 0)/(2 - 0) or c/(c + 1) = 1/3.The only solution is c = 1/2.