Analyzing concavity (graphical)
A function f of x is plotted below. Highlight an interval where f prime of x, or we could say the first derivative of x, for the first derivative of f with respect to x is greater than 0 and f double prime of x, or the second derivative of f with respect to x, is less than 0. So let's think about what they're saying. So we're looking for a place where the first derivative is greater than 0. That means that the slope of the tangent line is positive. That means that the function is increasing over that interval. So if we just think about it here, over this whole region right over here, the function is clearly decreasing. Then the slope becomes 0 right over here. And then the function starts increasing again, all the way until this point right over here. It hits 0. And then it goes, and the function starts decreasing. Just this first constraint right over here tells us it's going to be something in this interval right over there. And then they say where the second derivative is less than 0. So this means that the slope itself, whether it's positive or negative, that it's actually decreasing. We are going to be concave downwards right over here. The slope itself-- it could be positive. But it will be becoming less and less and less positive. And so we're looking for a place where the slope is positive, but it's becoming less and less and less positive. If you look over here, the slope is positive. But the slope is increasing. It's getting steeper and steeper and steeper as we go. And then, all of a sudden, it starts getting less steep, less steep, less steep, less steep all the way to when the slope gets back to 0. So if we want to select an interval, it would be this interval right over here. Our slope is positive. Our function is clearly increasing, but it is increasing at a lower and lower rate. So I will select that right over there. Let's do one more example. A function f of x is plotted below. Highlight an interval where f prime of x is greater than 0. So the same thing where our function is increasing, but it's increasing at a slower and slower rate. So our function is increasing in this whole region right over here, and we see it's really steep here, that it's getting less steep and less steep. And it's getting closer and closer to 0, the slope of the tangent line or the rate of increase of the function. So I would pick anything right around this region right here.