Executive summary1. Motivation and basic terminology1.1. Notion of constraints and objective function1.2. The extremes and the middle path1.3. Humbler matters: one-variable ambitions1.4. Geometrical and visual optimization1.5. Applications to real-world physical situations2. Important tricks in real-world problems2.1. The maximum is determined by the tightest constraint2.2. Some random tricks2.3. The intuition of tangency2.4. The heuristic of multiple uses2.5. Integer optimization3. Some notes from social and natural sciences3.1. An important maximization: Cobb-Douglas, fair share, and kinetics3.2. Frontier curves and optimal allocation3.3. Can spontaneous processes solve optimization problems?MAX-MIN PROBLEMSMATH 152, SECTION 55 (VIPUL NAIK)Corresponding material in the book: Section 4.5Difficulty level: Moderate to hard. This is material that you have probably seen at the AP level, butit is very important and there will be many additional subtleties that you may have glossed over earlier.What students should definitely get: The basic procedure for converting a verbal or real-worldoptimization problem into a mathematical problem se eking absolute maxima and absolute minima, solvingthat problem, and reinterpreting the solution in real-world terms.What students should hopefully get: Important facts about area-perimeter optima. The idea thatthe maximum is determined by the minimum, or most binding, constraint. The intuiton of tangency (asseen in the tapestry problem). The multiple use heuristic. The idea of transforming a function into anequivalent function that is easier to optimize. The procedure for and subtleties in integer optimization. Howsingle-variable optimization fits into the broader optimization context.Executive summaryWords...(1) In real-world situations, maximization and minimization problems typically involve multiple vari-ables, multiple constraints on those variables, and some objective function that needs to be maxi-mized or minimized.(2) The only thing we know to solve such problems is to reduce everything in terms of one variable.This is typically done by using up some of the constraints to express the other variables in terms ofthat variable.(3) The problem then typically boils down to a maximization/minimization problem of a function ina single variable over an interval. We use the usual techniques for understanding this function,determining the local extreme values, determining the endpoint extreme values, and determining theabsolute extreme values.Actions... (think of examples; also review the notes on max-min problems)(1) Extremes sometimes occur at endpoints but these endpoints could correspond to degenerate cases.For instance, of all the rectangles with given perimeter, the square has the maximum area, and theminimum occurs in the degenerate case of a rectangle where one side has length zero.(2) Some constraints on the variables we have are explicitly stated, while others are implicit. Implicitconstraints include such things as nonnegativity constraints. Some of these implicit constraints maybe on variables other than the single variable in terms of which we eventually write everything.(3) After we have obtained the objective function in terms of one variable, we are in a position to throwout the other variables. However, before doing so, it is necessary to translate all the constraints intoconstraints on the one variable that we now have.(4) When our intent is to maximize a function, it is sometimes useful to maximize an equivalent functionthat is easier to visualize or differentiate. For instance, to maximizepf(x) is equivalent to maxi-mizing f(x) if f (x) is nonnegative. With this way of thinking about equivalent functions, we canmake sure that the actual function that we differentiate is easy to differentiate. The main criterion isthat the two functions should rise and fall together. (Analogous observations apply for minimizing)Remember, however, that to calculate the value of the maximum/minimum, you should go back tothe original function.(5) Sometimes, there are other parameters in the maximization/minimization problem that are unknownconstants, and the final solution is expected to be in terms of those constants. In rare cases, thenature of the function, and hence the nature of maxima and minima, depends on whether those1constants fall in particular intervals. If you find this to be the case, go back to the original problemand see whether the real-world situation it came from con strains the constants to one of the intervals.(6) For some geometrical problems, the maximization/minimization can be done trigonometrically. Here,we make a clever choice of an angle that controls the shape of the figure and then use the trigonometricfunctions of that angle. This could provide alternate insight into maximization.Smart thoughts for smart people ...(1) Before getting started on the messy differentiation to find critical points, think about the constraintsand the endpoints. Is it obvious that the function will attain a minimum/maximum at one of theendpoints? What are the values of the function at the endpoints? (If no endpoints, take limitingvalues as you go in one direction of the domain). Is there an intuitive reason to b elieve that thefunction attains its optimal value somewhere in between rather than at an endpoint? Is there s omekind of trade-off to be made? Are there some things that can be said qualitatively about where thetrade-off is likely to occur?(2) Feel free to convert your function to an equivalent function such that the two functions rise and falltogether. This reduces the burden of messy expressions.(3) It is useful to remember the fact that the function xp(1 −x)qattains a local maximum at p/(p + q).That’s because this function appears in disguise all the time (e.g., maximizing area of rectangle w ithgiven perimeter, etc.)(4) A useful idea is that w hen dividing a resource into two competing uses, and one use is hands-downbetter than the other, the best use happens when the entire resource is devoted to the better use.However, the worst may well happen somewhere in between, because divided resources often performeven worse than resources devoted wholeheartedly to a bad use. This is seen in perimeter allocationto boundaries with the objective function being the total area, and area allocation to surfaces withthe objec tive function being the total volume.(5) When we want to maximize something subject to a collection of many