We know that the minimum is attained at a point . We only have to show that it can't be at or . If the minimum is attained at for example then exists such that if then . Now, don't forgive that is positive in order to find a contradiction.
Suppose is differentiable on [a,b] (with one sided derivatives at the end points). Show that if then the minimum of f is attained at a point . Note: You may not assume the derivative of f is continuous.
My first thoughts was just to use IVT on the derivative until I saw the note, now I'm not sure what to do. Any help where to start?