Consider the function f(x)=9x+4x^1. For this function there are four important i
ID: 2849000 • Letter: C
Question
Consider the function f(x)=9x+4x^1. For this function there are four important intervals: (,A], [A,B) (B,C], and [C,) where A, and C are the critical numbers and the function is not defined at B.
Find A
and B
and C
For each of the following intervals, tell whether f(x) is increasing (type in INC) or decreasing (type in DEC). (,A]:
[A,B):
(B,C]:
[C,)
Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x) is concave up (type in CU) or concave down (type in CD).
(,B):
(B,):
Explanation / Answer
f(x) = 9x + 4/x
critical points ==> f '(x) = 0 or f(x) is undefined
==> f(x) is undefined at x = 0
f '(x) = 0 => 9 - 4/x2 = 0
==> 4/x2 = 9
==> x2 = 4/9
==> x = 2/3 , -2/3
A = -2/3 , B = 0 , C = 2/3
(,A]: function is increasing, since for any value in that interval f '(x) > 0
[A,B): function is decreasing, since for any value in that interval f '(x) < 0
(B,C]: function is decreasing, since for any value in that interval f '(x) < 0
[C,) = function is increasing, since for any value in that interval f '(x) > 0
(,B): Concave up , since f '(x) increases as x increases in the interval
(B,): Concave up , since f '(x) increases as x increases in the interval