E XTREME V ALUE T HEOREM Theorem 121 Extreme Value Theorem If y f x is

E xtreme v alue t heorem theorem 121 extreme value

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E XTREME V ALUE T HEOREM Theorem 12.1. (Extreme Value Theorem) If y = f ( x ) is continuous on the closed interval [ a, b ] , then, restricted to the interval [ a, b ] , f has an absolute maximum and an absolute minimum on [ a, b ] . 2 f (c) Jfk ) f (c) Effy for all X " near " C for all X " near " C Linan open interval on either side ofc ) ( in an open interval on either side of c) y=x2H LOCAL MIN VALUE is 1 < 2 It's also the GLOBAL MIN VALUE EX LOCAL MAX ^ ix. . " ! : " naeizte.r.EE#*sreehoEaploFnxtnem- v LOCAL MIN @ X=2 Ex . a#y=rx GLOBAL MIN VALUE ISO , at X= O , but it is not a LOCAL MIN < & BECAUSE TX is UNDEFINED FROM LEFT OF X=O = # top Ofa " hill " * if f' G) DNE , then f could still have A bo¥ot a' ' valley " iatwooutebminafmaaxorantr bust off
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Note. If f is not continuous on [ a, b ] , then the Extreme Value Theorem is not applicable. Even if f is continuous, if the interval is not closed, then the Extreme Value Theorem is not applicable. Strategy to find the absolute extrema of a continuous function f ( x ) on a closed interval [ a, b ] 1. Find the critical numbers of f . 2. For each critical number c such that c 2 [ a, b ] , compute its value f ( c ) . 3. For the endpoints x = a and x = b of [ a, b ] , compute the values f ( a ) and f ( b ) . 4. The absolute maximum value of f on [ a, b ] is largest value computed in steps 2 and 3. 5. The absolute minimum value of f on [ a, b ] is smallest value computed in steps 2 and 3. Example 12.2. Find the extreme values of g ( t ) = 2 t 10 (4 - t 2 ) 5 on the closed interval [ - 1 , 2] . 3 = ^ ^ I . ÷ . . . I . ' in . f v v this function is NIT Continuous On Cab ] this function is continuous on Laib ) It has he GLOBAL EXTREMA on laid but the interval is Net closed . It has he GLOBAL EXTREMA on Carb ) . gYH=2ot9( 4- t45t2t ' 45114 - tY4f2t ) same domain as g C all real # s ) ooo no type 2 critical # s .
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