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# Class Note for MATH 1314 at UH 2

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Date Created: 02/06/15

M1314 Lesson 13 1 Math 1314 Lesson 13 Absolute Extrema In earlier sections you learned how to find 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 find absolute extrema that is the highest high andor the lowest low on the domain of the function or on a specific closed interval Absolute Extrema on the Domain of f Definition If fx 5 fc for all xin the domain of I then fc is called the absolute maximum value of f If fx 2 fc for all x in the domain of I then fc is called the absolute minimum value of f Sometimes you will be asked to find the absolute extrema over the interval 0000 Example 1 State the absolute maximum andor absolute minimum values v 141950105111 Mn X30 90 0 Abso uki YVYtrx 050 79277757574737271 1234557291 M1314 Lesson 13 2 Example 2 State the absolute maximum andor absolute minimum values AbSOlq LQ Xvyz 46 22 1019501th MM is 2 As you can see from these two examples the absolute extrema may or may not exist To find absolute extrema on oooo 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 fis continuous on a closed interval a b then fhas both an absolute maximum value and an absolute minimum value on a b M1314 Lesson 13 3 Finding the Absolute Extrema of fon a Closed Interval 1 Find the critical points of fthat lie in a b 2 Compute the value of the function at every critical point found in step 1 and also compute 1a and Kb 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 yvalue of the point on the graph or t Read the question carefully 2 Example 3 Find the absolute extrema of fx x on the interval 1 3 F x72x 0 2x20 gtO tes gtlt J XoXT 3 Flt lt7 0 021 423 032 0 43 cz q meme max is 1 bolu klt Min is 0 Example 4 Find the absolute extrema of fx x3 3x2 1 on the interval 2 2 CDPltx73gtltLCagtlt 0 Absoh x Max 3951 3XCXZUO A bsoh rL m7 5quot 3x10 XZO x20 xL Eest gtltzfgtltogtlt2 4307 6233136014 2393 439 o CO33sz 40 2 5Lquotlr 32 c M1314 Lesson 13 4 Example 5 Find the absolute maximum value and the absolute minimum value of the function fx x2 2x over the interval 3 1 5900quot 2X4zo Absolu e maxf53 axt39z absolu e Min 5 X734 Q X393 X 1K 1 40337 3 zC337 q quot3 Cagtrc 3l260quot32 Cm 0 107 39z3 Example 6 Find the absolute maximum value and the absolute minimum value of the function fx 3x4 4x3 on the interval 1 2 a 4300 IZX3 080 Ab56QSltZMQX l5 1 ZXI CX D 0 Absehu cz mm 13quot 12x1 0 xo Xfo X x XOxxz 3 47 44 300 HQ D 3 5r p o 3 gt 39 Meg s at 07 3C0 LGC03 39 0 413 32 i 42 3 Ll quot 31 Example 7 Find the xcoordinate of the absolute extrema of 2 the fundio f0 3x5 2x on the interval 1 1 F x3lt3XFV339Z xz I XODX 4 z 0 VCD 360quot 26 x 9 Z Z E 3 2773S Ida E21 725 Y3 7 35 39ZHD39Q ch is und ww Z gtltYB 3 I rZlt l attxjjzst X Absoh ct max is 5 Wm JAI VotlkAQ AbSGlm CQ mlh 15 O M1314 Lesson 13 5 Example 8 An apartment complex has 100 onebedroom apartment units Research shows that the monthly profit 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 profit What is that monthly profit 39 loo PIX 24gtlt 2zSa to gt 2125 qt 2 Test X70 X3qul X ioo 48 000 1 12390 48000 r LII Ecaqi zzsquKD qxlouo 53032 PCroo 391 l ojL39l39ZZSVCldovtrglooo 57000 maximum PWO39FIL is oi 59032 q apartments ranted 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 Cx 00025x2 80x 10000 Find the average cost function Find the level of production that results in the smallest test to verify that you have found a minimum cost I my 3 C04 2 OOZSKZJ w 41 oozsx ol lwox D x K v 39 39 393 C39Cx I 2 J IO3929o Cquot OIOUO ZX XL Q 3 ooz 194 zopwx X T X1 942 C f2 3 39 oo 2 1 OOOCZIOGQ x1 llo 70 gel a I l K zlmo C lCZmoo M 395 20360 2089 From this section you should be able to T lt0 Identify absolute extrema on 00 00 from a graph of 2 max Find absolute extrema on 5 b algebraically Solve word problems having to do with absolute extrema C 2 3067 2 5S0 CeS l P2r um H

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