Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
3
Academic year
2022
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71
Review critical points A critical point is a point where the partial derivatives are both 0. In this lecture, wewill continue to explore critical points and learn how to actually decide whether aparticular point is a minimum, maximum, or saddle point. There are various kinds of critical points. Local minima, local maxima, which are likethat, and saddle points which are neither minima nor maxima. And, of course, if youhave a real function, then it will be more complicated. It will have several criticalpoints. In this example, there are two maxima and a saddle point in between them. Thecontour plot shows the maxima as circles that narrow down and shrink to theirhighest points. The saddle point appears as a figure eight-shaped level curve thatcrosses itself.And if you move up or down here, along the y direction, the values of a function willdecrease; along the x direction, the values will increase. So you can see where thecritical points are just by looking either at the graph or contour plot.
So the question is, how do we decide between various possibilities? Local minimum,local maximum, or saddle point. How do we find the global minimum or maximum ofa function? To decide where a function is the largest, generally you must compare its values. Forexample, here if you want to know what is the maximum of this function, well wehave two obvious candidates. We have this local maximum and that local maximum.And the question is which one is the higher of the two? In general, you find the maximum of a function by computing the function at bothpoints and comparing the values. If you know that it is 3 at one of them and 4 atanother, then 4 wins. In this case, though, both critical points are tied for maximum. Ifyou are looking for a minimum of this function, then it is not going to occur at any ofthe critical points.So where is the minimum? It turns out that the minimum is actually on the boundary,or at infinity. Global minima and maxima can occur off critical points. We have to
check for boundary and infinity behavior to know where a minimum or maximum willactually be. So in general, this could occur either at the critical point or on the boundary of adomain of definition that we are considering. We'll get back to that; for now let's try tofocus on the question of what type of critical point it is.
Partial Derivatives in Action: Tangent Approximation Explored
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