How to solve Max/Min problems, using Lagrange multipiers, Part I

Many of the steps are the same as for ordinary max-min problems!

Step 1: Read the problem carefully. It's useful to underline any numbers of formulas in the problem.

Step 2: Answer the three questions.

• What quantity are we trying to maximize or minimize?
• What variables does the quantity depend on?
• What restrictions are there on the variables? What relations are there between the variables?

Step 3: Find a formula relating the quantity we are trying to optimize to the variables that it depends on. (For geometric problems, a diagram is often helpful.)

Step 4: Write the relations you found in step 2 in the form g(x,y)=c for some function g and some constant c. Here you want to re-write the relation so that it is the equation of a level curve or level surface.

Step 5: Use the method of Lagrange mulitpliers to find the critical points. (How do I do that?)

Step 6: Look over the critical points and the behavior of the function from step 3 on the boundary of its domain, to decide where the max or min occurs. In this step, if their are several critical points, you may want to plug them into the function. If the level curve or surface that you are working on is unbounded, then you may want to look at the formula for the function to decide if the function continues to increase or decrease as the variables go to infinity.

Return to Quiz 7.