Now we're going to look a little bit more at these particular functions that we just had. The V(x) which was, let's see what was it? It was x times (10 -2x) times (15 -2x), which was 150x -100x squared plus 4x cubed. But that's right, 30, 40, 50, 30,40,50 sorry, that was 50. 50, 30, 150, here we go. That was the V of x, and our V prime of x was 150 minus 100x plus 12x squared. Now I graphed this one for you earlier, right? We sketched that graph and it looked something like this. That was V(x), and now I could also sketch this one on top of it which would come and look something like, let's see what is it doing? Here, the function is increasing, so my derivative is positive. So it does something like it hits zero about where that one goes and it's going to hit zero again there. So it'll look something like that, okay? That's my V prime of x, okay? Now, if I didn't want to graph these, how would I know whether I've had a maximum or minimum, right? I said we knew that 1.96 gave me the maximum, right? Because I had already graphed V of x to start with. What if my function is so complicated? I don't know how to graph it and I can't put it in whatever kind of graphing tool or not trusting whatever kind of graphic tools. Remember we talked before about the role of the second derivative, right? And telling us whether something was concave up or concave down. So I know that I had V prime was zero at x approximately equal 1.96 or what was the other one? 6.37, okay, so if I want to know which of those was a maximum, which was a minimum, I could look at the second derivative. So let's look at the second derivative of my volume function. Let's see derivative of 150, derivative of minus 100 x is minus 100 plus 24x, right? To bring the two down, multiply, reduce that by one. So at around two, right, that's pretty close to two. At around two, this is going to be minus 100 plus 48, it's going to be negative. So, I know that V double prime at 1.96 is negative. What does that mean? That means it's concave down, right? Which means I had a maximum, right? If it's concave down it, doesn't hold water, which means it was a maximum. So that tells me that 1.96 is a relative max. And how about at 6.37? All right, the second derivative at 6.37. This would be minus130 something, it would be 130 something, -100 would be positive. So that tells me that 6.37 would give me a relative, min. Okay, so that's how we can use the second derivative without ever having to graph the function. To tell whether what we got when we solve for the first derivative equal to zero would have given us a maximum or a minimum, right? So, there's two ways you can do it, you can graph it and tell from that. Or you can take the second derivative and use, it's called the second derivative rule to see whether it was a maximum or minimum.