Optimization problems ask you to find the largest or smallest value of a quantity. The key idea is to turn the real situation into a function and then look for its extreme values.
Decide what must be maximized or minimized, such as area, cost, distance, or time. Write it as a variable or expression.
Translate the information in the problem into equations. Often, one variable can be expressed in terms of another, so the target quantity becomes a function of a single variable.
Substitute the constraint into the quantity to optimize. Simplify carefully so the final expression is as simple as possible.
Differentiate the function, then solve where the derivative is zero or undefined in the allowed domain. These are the main candidates for an optimum.
Use a sign chart, a second-derivative test, or compare endpoint values when they exist. This tells you whether the point is a maximum or minimum.
State the value clearly and simplify exactly if possible.
Make sure your answer fits the original conditions and has the correct units. If the problem asks for a maximum, your result should be larger than nearby values; for a minimum, it should be smaller.
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