Question: Let f and g be continuous functions on [a, b] such that f(a) g(a) and f(b) g(b). Prove that f(x_o) = g(x_o) for at least one x_o in [a, b].

Hint: Let h = f - g. Then h is continuous and h(b) 0 h(a). Now apply the Intermediate Value theorem which states that

If f is a continuous real-valued function on the interval I, then f has the intermediate value property on I. Whenever a, b I, a < b and y lies between f(a) and f(b) [i.e, f(a) < y < f(b) or f(b) < y < f(a)], there exists at least one x (a, b) such that f(x) = y.