I'm looking at problem 42 in Chapter 3 of Folland's Real Analysis text, which asks to prove that if F: (a,b) -> R is a convex function and t_o is in (a,b), then there exists a real number b such that F(t)-F(t_0) ≥ b(t-t_0) for all t in (a,b).

I thought about invoking the Lipschitz condition, but that doesn't seem to get me all the way there since F is absolutely continuous on every compact subinterval of (a,b). Can anyone help me out?