Screen Shot 2019-11-30 at 11.11.37 PM.png - In chapter 5 your textbook refers to four"tests Test for intervals where f(x is increasing and decreasing

Screen Shot 2019-11-30 at 11.11.37 PM.png - In chapter 5...

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Unformatted text preview: In chapter 5, your textbook refers to four "tests": Test for intervals where f(x) is increasing and decreasing, Test for Concavity, First Derivative Test, and Second Derivative Test. Note that the first two tests can be summarized using a sign chart. Both the First and the Second Derivative TESTS determine whether a critical point, c, is a relative extremum (max or min). But even though both tests determine relative extrema, they do so differently. The First Derivative Test allows you to locate the relative extrema by analyzing the increasing/decreasing behavior of the graph to the left and to the right of the critical number (increasing on the left and decreasing on the right indicates a relative maximum, for example). To determine whether a critical number c is a relative extremum using the Second Derivative Test, you would evaluate the second derivative at c and determine the concavity at that point (positive indicates concave up and thus a relative minimum, negative indicates concave down and thus a relative maximum, zero indicates the test is inconclusive). 8. Let f(x) =9x+ -. Find the critical number(s) and determine whether they are local extremum/extrema using both the First Derivative and the Second Derivate Tests. f(x ) = 9* + - fi(X ) = 9 -1 X z 9 x 2 ( x - 3 ) ( *.+ # ) critical points: x = + 1/ 3 local max : x = -1/3 I " ( = ) = - 54<D -1/ 3 1/ 3 10calming x = 1/3 * " ( 4/3 ) = 54- 0 9. Let f (x) = x + - 15 -x2 - 18x-1 0 = * 3 + 2 X - 18 x - 1 x 2 -6 ( - 6 , 1 a ) Find f'(x) and f" (x). fi( X ) = 3 x2 + 15. 2X - 18:1 +0 - = 3x2+ 15x- 18 -1 f (- 6 5 f ( 1 ) 5" ( x ) = 6 * + 15 3 ( x 2 + 5 X - 6) - 3 ( X + 6) (x-1) X = - tix= 1 b) Construct a first derivative sign chart to determine the intervals where f is increasing or decreasing. Increasing: (-00, - 6) (1 00) - 6 O - min max f' ( - 8) = 59 # S' (2 ) =- 9.75 Decreasing: ( - 6,1 ) f'( - 2) = - 36 4 f' ( 2 ) = 24 * Use the First Derivative Test to find all relative maxima and minima. ( list the coordinates) Rel. max : X= - 6 Rel. min : x = 1 5 of 7 2 x 2 + 15x-18 - 3( X+6 ) ( x- 1 )...
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