lecture26 - ( x ) < 0 on-∞ < x < 1. Application:...

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Math 006 (Lecture 26) Analyzing Graphs Here are the general patterns for analyzing Graph: (1) f 0 ( x ) > 0 and f 00 ( x ) > 0 (2) f 0 ( x ) > 0 and f 00 ( x ) < 0 (3) f 0 ( x ) < 0 and f 00 ( x ) > 0 (4) f 0 ( x ) < 0 and f 00 ( x ) < 0 Procedure for graphing a polynomial 1. Analyze f ( x ) Find the x -intercept(s) and y -intercept of f ( x ). 2. Analyze f 0 ( x ) Find the critical point(s) of f ( x ). Construct a sign chart for f 0 ( x ), determine the interval(s) where f ( x ) is increasing and decreasing, and find local maxima and minima. 3. Analyze f 00 ( x ) Find the inflection point(s) of f ( x ). Construct a sign chart for f 00 ( x ), determine the interval(s) where f ( x ) is concave upward and concave downward. Example 1. Graph y = x 4 + 4 x 3 . 1
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Example 2. Graph the function f , if f satisfies (a) f (0) = 2, f (1) = 0, f (2) = - 2; (b) f 0 (0) = f 0 (2) = 0; f 0 ( x ) > 0 on -∞ < x < 0 and 2 < x < ; f 0 ( x ) < 0 on 0 < x < 2; (c) f 00 (1) = 0; f 00 ( x ) > 0 on 1 < x < ; f 00
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Unformatted text preview: ( x ) < 0 on-∞ < x < 1. Application: Point of diminishing return The value of x where the rate of change of f ( x ) changes from increasing to decreasing is called the point of diminishing return . Example 3. A discount appliance store is selling 200 large-screen TV sets monthly. If the store invests $x thousand in an advertising campaign and the ad company estimates that sales will increase to N ( x ) = 4 x 3-. 25 x 4 + 500 , ≤ x ≤ 12 . When is rate of change of sales increasing and when is it decreasing? What is the point of diminishing returns and the maximum rate of change of sales? 2...
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This note was uploaded on 09/04/2010 for the course MATH MATH006 taught by Professor Forgot during the Fall '09 term at HKUST.

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lecture26 - ( x ) < 0 on-∞ < x < 1. Application:...

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