Let h(x)=f(x)-cx, then h' (x) = f' (x)-C. Because f' (a) < c < f' (b), there is h' (a) < 0 < h' (b).

Because H' (a) < 0, it means that h(a) can't be the minimum value of h(x) on [a, b], and H' (b) > 0, it means that h(b) can't be the minimum value of h(x) on [a, b], so h(x) must be in the interior point ξ.