Step By Step Calculus » 13.6 - Optimization Problems

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Guidelines for Solving Optimization Problems:
Step 1 (Understand the Problem) Read the problem carefully until you get a clear understanding of it. A rule of thumb is to read the problem at least three times before trying to solve it. Answer the following questions when reading: (1) What are the unknowns? (2) What are the knowns? (3) What needs to be maximized/minimized? (4) What are the constraints?
Step 2 (Sketch) If appropriate, sketch a diagram and identify the quantities of interest on it.
Step 3 (Introduce Notation) Denote the unknowns with symbols and label the diagram with these symbols.
Step 4 (Formalize the Problem) Write the function which you are asked to maximize or minimize. Call this function as the “optimization” function. Then, write down all the constraint equations related to the problem. As in the previous example, MOST optimization problems will have one “optimization” function and one “constraint” equation. However, some problems may have no constraint equation and some may have two or more constraint equations.
Step 5 (Convert Optimization Function) Express “optimization” function in terms of a single variable with the help of “constraint” equations.
Step 6 (Optimize) Use the techniques (i.e. the first or second derivative test) learnt in previous section to find a maximum or minimum value of the “optimization” function. This will lead to the value of unknowns.