Let f be continuous on [a, b] and assume the second derivative f exists on (a, b).

Let f be continuous on [a, b] and assume the second derivative fʹʹ exists on (a, b). Suppose that the graph of f and the line segment joining the points (a, f(a)) and (b, f(b)) intersect at a point (x0; f(x0)) where a < x0 < b. Show that there exists a point c ∈ (a, b) such that fʹʹ(c) = 0.