The derivative is increasing on that interval, which means that the original function is concave up. F is positive on that interval. That's not sufficient, that's not a sufficient calculus-based justification, because if your derivative is positive, that just means your original function is increasing, it doesn't tell you that your original function is concave up. F is concave up on the interval. Well, just because your derivative is concave up doesn't mean that your original function is concave up. In fact, you could have a situation like this where you're concave up over that interval, but for much of that interval right over here, if this was our graph of f or g prime, we are decreasing, and if we're decreasing over much of that interval, then actually on this part, our original function would be concave down. The graph of g has a cup u shape on the interval.