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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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- Fall '19