So I have the function g here, it's expressed as a fourth degree polynomial, and I want to think about the intervals over which g is either concave upwards or concave downwards. Now let's just remind ourselves what these things look like. So concave, concave upwards, is an interval, an interval where you're concave upwards, is an interval over which the slope is increasing, and it tends to look like an upward opening u like that, and you can see here that the slope over here is negative, and then as x increases it becomes less negative, it actually approaches zero, it becomes zero, then it crosses zero, it becomes slightly positive, more positive, even more positive, so you can see the slope is constantly increasing. And if you think about it in terms of derivatives, it means that your first derivative is increasing over that interval, and in order for your first derivative to be increasing over that interval, your second derivative, f prime prime of x, actually let me write it as g, because we're using g in this example, in order for your first derivative to be increasing, your, let me write this, so g, so concave upward means that your first derivative increasing, increasing, which means, which means that your second derivative is greater than zero. And concave downward is the opposite. Concave downward, downward, is an interval, or you're going to be concave downward over an interval when your slope is decreasing. So g prime of x is decreasing, or we can say that our second derivative, our second derivative is less than zero.