Where is the maximum? In order to get bigger, one must increase the size of the function. To get smaller, onemust reduce the size of the function. For example, when we see a graph increasingin height, it means that the function is getting larger. It's called visualizing. Let us draw a circle to represent the situation, as shown below. When we see the function's level curve and want to visualize what the graph lookslike, we remember that the level curve is perpendicular to the gradient. The gradientis the direction of steepest increase, so what helps us do that is to take our piece ofcardboard—we use it to visualize graphs—and stick it on the board so that its bottomis perpendicular to that vector. So this is what the function looks like near here. Then our question became, if we follow this vector field, are we going uphill ordownhill? So this vector field is a function. You can see that I'm going uphill. We`renot going the steepest possible direction uphill but going uphill. As you do yourproblem sets or whatever you can have your ID handy in one hand. And when yousee this vector field making such a graph then put it down there on the board andsay that's what the graph looks like.
So now that we've discussed the concept of graphing functions, let's apply thatknowledge to a problem. The question is, where in this picture is the maximum of thefunction f?