Let be continuous on [a, b] and differentiable on (a, b). Also, suppose that (a) = (b) and that c is a real number

Let ƒ be continuous on [a, b] and differentiable on (a, b). Also, suppose that ƒ(a) = ƒ(b) and that c is a real number in the interval such that ƒ'(c) = 0. Find an interval for the function g over which Rolle's Theorem can be applied, and find the corresponding critical number of g (k is a constant).

(a) g(x) = ƒ(x) + k 

(b) g(x) = ƒ(x - k)

(c) g(x) = ƒ(kx)