Now if we want to optimize this profit function analytically, the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points? And if one of them is a maximum point, then we can say, well, let's produce that many. That is going to be, we will have optimized or we'll figure out the quantity we need to produce in order to optimize our profit. So to figure out critical points, we essentially have to find the derivative of our function and figure out when does that derivative equal zero or when is that derivative undefined? That's the definition of critical points. So p prime of x is going to be equal to negative 3x squared plus 12x minus five. And so this thing is going to be defined for all x.