Define a new function g on the interval [a,b] by Then g is also continuous on [a,b] and differentiable on (a,b). If there are two distinct points in [a,b] with then , as required.
Suppose that there do not exist two such points, so that g takes each of its values only once. In other words, g is an injective function on the interval [a,b]. Use the intermediate value theorem to deduce that g must be a monotone function on the interval. It then follows that g'(x) must always have the same sign (either for all x in [a,b], or for all x in [a,b]). But , which means that g' has either a maximum or a minimum at c, and hence so does f' (because g' and f' only differ by a constant).
Taking the contrapositive, it follows that if f' does not have a max or a min at c, then points with the desired property must exist.