Optimization Using Derivatives

    In this topic we give a few examples on how to set up a function to be optimized using its derivative. In general, the first step in solving an application problem is to understand the problem; maybe ask what are the unknowns?, and what are the given quantities? Then the next best step is usually to draw a picture, labeling the unknowns and introducing notation. The final step should always be to check the solution to see that it makes sense for the given questions.

Example (Optimizing with Area) Someone with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?

    Solution. Let be the lengths of the 2 sides and let be the lengths of the other 5 sides. We have because there are three divides making the four pens. The area of the four pens is thus we can solve for obtaining So a function of the area is



We find and so the critical number is The largest possible area is

Example (Optimizing with Geometry) Find all points on the circle such that the product of the -coordinate and the -coordinate is as large as possible.

    Solution. In the first quadrant we have and so we want to maximize We compute, Thus the critical number is The maxium value is  



Thus the points are and

Example (Optimizing with Numbers) Find two nonnegative numbers whose sum is and the product of whose squares is as large as possible.  

    Solution. We are looking for two nonnegative numbers, say and with and We want to maximize We compute,
    


Thus the critical numbers are We find and the largest possible value to be with   

Example (Optimizing with Numbers)  Under the condition that minimize when and

    Solution. We want to minimize We compute,
    


Thus the critical numbers are and . We find and the smallest possible value to be Thus the values are and

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Optimization Using Derivatives
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/optimization-using-derivatives.html