Constrained optimization: Big picture If we have a function of (x, y) and a bounded region R, then there are twopossibilities: Either the maximum occurs at a critical point, meaning that the xderivative at this point equals 0 and the y derivative equals 0. Or another way ofsaying it is that the gradient at this point equals 0. One possibility is that the maximum occurs at a critical point. This is true when thegradient vanishes, which means that both the derivative of x and the derivative of yvanish. Another possibility is that the maximum occurs at a boundary point. Perhaps it's helpful to observe that the function may be visualized as a graph in the xand y plane, where it looks like an upside-down bowl.A point (x0, y0) is called a critical point of a function f(x, y) if f(x0, y0) = 0. Themaximum of the function is at the top of the bowl. Example 2: Let R be the region defined by y = f(x). Suppose the maximum value of foccurs along an edge of R, as shown above. If f has a slope at some point on thisedge other than where it is equal to zero (a critical point), the slope will be negativeat that point. What if in this first example of the bowl, instead of being rotated downwards, it wasrotated upwards? What would be the maximum then?
If we have an upward bowl, then the minimum will be somewhere in the middle. Thisis a critical point. The maximum will be along the edge. If the bowl is perfectlysymmetric, then there could be a tie. So how can we choose x and y to make f bigenough? There could be a tie. There could be several different, maybe even infinitelydifferent, many different x's and y's that give the same value,which is the biggestvalue. That big value is the maximum. It occurs at all of those points so it could occurat all of the points all the way around the edge. To find the maximum, you take your function and its interval, find the critical pointsinside that interval, and check the boundary of those points. If there are two criticalpoints on a boundary of an interval, then you have found your maximum. Well, let's do the same thing. We can find all of the critical points. There are typicallya few critical points in the domain. Then we have to check the boundary. The biggestdifference is that checking the boundary is much more complicated because it's acurve and there are infinitely many points on it. So we have to really think about howto handle it.