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# Class Note for MATH 1314 with Professor Marks at UH

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This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 20 views.

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Date Created: 02/06/15

Math 1314 Lesson 13 Absolute Extrema In earlier sections you learned how to nd relative local extrema These points were the high points and low points relative to the other points around them In this section you Will learn how to nd absolute extrema that is the highest high andor the lowest low on the domain of the function or on a speci c closed interval Absolute Extrema 0n the Domain of f De nition If f x S f c for all x in the domain off then fc is called the absolute maximum value off If fx 2 fc for all absolute minimum value of x in the domain off thenfc is called the Sometimes you Will be asked to nd the absolute extrema over the interval ltgtltgt 00 Example 1 State the absolute maximum andor absolute minimum values 1234567891 Example 2 State the absolute maximum andor absolute minimum values y 79 72 77 76 is 74 7 72 4 1234557291 As you can see from these two examples the absolute extrema may or may not exist To nd absolute extrema on 0 00 or on the entire domain of the given function algebraically you must graph the function using the guide to curve sketching Absolute Extrema on a Closed Interval More often you will be asked to find the absolute extrema over a closed interval a b and for a continuous function those will always exist and they are much easier to find Theorem If a function f is continuous on a closed interval a b then f has both an absolute maximum value and an absolute minimum value on a b Finding the Absolute Extrema of f on a Closed Interval 1 Find the critical points of f that lie in a b 2 Compute the value of the function at every critical point found in step 1 and also compute at and fb 3 The absolute maximum value will be the largest value found in step 2 and the absolute minimum value will be the smallest value found in step 2 Note The absolute maximum value or the absolute minimum value refers to the y value of the point on the graph or x Read the question carefully Example 3 Find the absolute extrema of fx x2 on the interval 1 3 Example 4 Find the absolute extrema of fx x3 3x2 1 on the interval 2 2 Example 5 Find the absolute maximum value and the absolute minimum value of the function fx x2 2x over the interval 3 l Example 6 Find the absolute maximum value and the absolute minimum value of the function f x 3x4 4x3 on the interval 1 2 Example 7 Find the x coordinate of the absolute extrema of the function 2 fx 3x3 2x on the interval 1 l Example 8 An apartment complex has 100 onebedroom apartment units Research shows that the monthly pro t in dollars realized from renting out x apartments is given by Px 12x2 2256x 48000 How many units should be rented out to maximize the monthly pro t What is that monthly pro t Example 9 A company that produces digital cameras wants to minimize its production costs They estimate that their total monthly cost for producing the camera is given by Px 00025x2 80x 10000 Find the average cost function Find the level of production that results in the smallest average production cost Use the second derivative test to verify that you have found a minimum cost From this section you should be able to Identify absolute extrema on 00 00 from a graph off Find absolute extrema on 61 b algebraically Solve word problems having to do with absolute extrema

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