3.2.notes - = − (+)( − ) (+) = − (+)( − ) (+) = −...

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MAC 2233/001-006 Business Calculus, Fall 2011 CRN: 81700–81702, 81705, 81716, 81715 Assistants : Matthew Fleeman, Arbin Rai, Tadesse Zerihun, Vindya Kumari, Junyi Tu, Jing- han Meng An example on relative extrema Example . (Similar to #39 in § 3.2) Determine all relative extrema of the function: f ( x ) = x 3 x 2 1 . Solution . Note the the ±rst 3 steps are about monotone properties of f ,a dn are therefore te same as those in the example for § 3.1. Step 1 . The derivative is f ( x ) = (3 x 2 )( x 2 1) ( x 3 )(2 x ) ( x 2 1) 2 = x 4 3 x 2 ( x 2 1) 2 = x 2 ( x 2 3) ( x 2 1) 2 . Step 2 . (a) When f ( x ) = 0, we have x 2 ( x 2 3) = 0, giving that x = 0 , ± 3. (b) When f ( x ) is unde±ned, we have x 2 1 = 0, giving that x = ± 1. (NOTE: Only x = 0 , ± 3 are critical numbers because f ( x ) is not de±ned at x = ± 1.) Step 3 . intervals ( −∞ , 3) ( 3 , 1) ( 1 , 0) (0 , 1) (1 , 3) ( 3 , ) test values 2 1 . 5 0 . 5 0 . 5 1 . 5 2 sign of f (+)(+) (+) = + (+)( ) (+) = (+)( ) (+)
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Unformatted text preview: = − (+)( − ) (+) = − (+)( − ) (+) = − (+)(+) (+) = + conclusion ր ց ց ց ց ր Step 4 . Since (from the arrows in the table) it is possible to have relative extrema at x = ± √ 3 only, the candidates of interest are x = ± √ 3. (a) We need to check if they are critical numbers. That is, we need to check if f ( x ) is de±ned at these numbers. Indeed they are (because f ( x ) is not de±ned only when the denominator is 0, that is, at x = ± 1. (b) Using the ±rst derivative test, we have a relative maximum at x = − √ 3, with f ( x ) = − 3 √ 3 2 ≈ 2 . 5981, and a relative minimum at x = √ 3, with f ( x ) = 3 √ 3 2 ≈ 2 . 5981. We get the following if we graph the function:...
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This note was uploaded on 01/02/2012 for the course MAC 2233 taught by Professor Danielyan during the Fall '08 term at University of South Florida.

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3.2.notes - = − (+)( − ) (+) = − (+)( − ) (+) = −...

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