Critical point practice. Worked examples We'll now consider a contour plot, which is given to us as follows. The values are notindicated. We want to identify a unique saddle point on this contour plot and label itpoint A. There are two points that are either maximums or minimums but we cannottell because the labels are not on this contour plot. However, for convenience, we will label the points in the following manner: B and C.The points B and C are either maxima or minima, but we can still find them andidentify them. The second part presents two scenarios. First, although we do notknow the values for the graph, we can consider possible configurations that couldyield both B and C being maximal. Second, we can consider another scenario inwhich B is maximal but C is minimal. To get started, let's answer the first question by writing the points on our originalgraph. When we are looking for a minimum or maximum on a contour plot, weshould keep in mind that a minimum or maximum will always be contained inconcentric contours that are either approaching the maximum from below or theminimum from above. If we look at the graph, we see that as the radius of the circleincreases, the number of rings around it also increases. Somewhere in betweenthese two extremes, there must be either a maximum or minimum because thegraph never passes through another contour plot; it's either a maximum or minimumof the function inside this innermost ring. Similarly, there must be a minimum ormaximum for this function somewhere outside the innermost ring. So let's call this B and call this C. Now, we also have a saddle point A in thisproblem. And it's a little hard to see in the contour plot, but basically what washappening is that you have these contours (so that contour and that contour are thesame). So the value of the function here and here are the same. And yet if we look atthis point and this point, they either go up or down. Let's assume that the values go up. In this direction, the values are determined by acontour curve. So we have a saddle point A in the middle there. It's going to be whenyou have two either maxima or minima rising out and you have a contour, which iscontaining the point in the middle. So those are our points A, B, and C that we'regoing to be interested in. So now the second question that we have to consider is, can B and C both bemaximal? And then the third question is, can B be maximal and C minimal? We can demonstrate that B and C are both maximal by sketching an example.
Here is a graph of this function in three dimensions. If we want B and C to both be attheir maxima, then let me draw the contour lines. First, we have this one. First of all,we had this one, then we had another one, then we had a peak, then we had a peak.So if we want to draw this in three dimensions, then what we just need to do is followthese contour plots up out of the plane and into space. So this goes up and thenthere’s a maximum and then it comes back down along the contour lines and then itgoes back up and then it goes back down. We need to flesh out those curves so thatthey can connect with one another across the surface of the graph. And so indeed, we do see that it's possible for both B and C to be maximal. Here isan example of such a thing. And as promised, it is much clearer now how Abecomes a saddle point because you have these two mountains rising up, and thevalley in between them is necessarily a little saddle here. It increases in thisdirection, and it decreases in this direction.Now, for the second problem, we are asked whether B can be maximized and yet Cminimized. The answer is still yes. To think about this graphically, imagine that we
start over here and we dig a hole and as we're digging we throw the dirt over itbehind us. So we'll have a hole, a dip here and then we'll pile that entire hole up overhere. Now, notice that both of these functions have the same contour plot. The concentricrings on C indicate that the function is increasing. They are the same below, becausethis is essentially the same thing. The concentric rings on this surface show that B is a maximum point, while the samerings indicate C is a minimum point. This illustrates that contour plots do not alwaysreveal the global behavior of a function unless we label the values at each contour.Notice that in both cases A is a saddle point; it is increasing in one direction anddecreasing in the other.