The function f(x) = x^2/(1 + x^4) has ___________Complete the sentence with one of the choices listed below.

Question

A. exactly two inflection points, one local maximum and no local minimum
B. exactly three inflection points, one local maximum and one local minimum
C. exactly four inflection points, two local maxima and one local minimum
D. no inflection points, no local maximum and no local minimum
E. properties not described above

Answer 1

C
Use the quotient rule to find f'
f'(x) = [(1+x^4)(2x) - (x^2)(4x^3)]/(1+x^4)^2
= [2x + 2x^5 - 4x^5]/(1+x^4)^2
= [2x - 2x^5]/(1+x^4)^2
= (2x)(1-x^4)/(1+x^4)^2
1+x^4 never equals 0.
2x(1-x^4) = 0 when
2x = 0 or 1-x^4 = 0
x = 0, x = -1, or x = 1
You have 4 intervals:
(-infinity,-1), (-1,0), (0,1) and (1,infinity)
Pick a test point to find out whether f'(x) is negative or positive.
(-infinity,-1) is positive
(-1,0) is negative
(0,1) is positive
(1,infinity) is negative
This means that x = -1 is a local max, x = 0 is a local min and x = 1 is a local max.
That is, there are two local maxima and one local minimum
C is the only choice. (so you do not even have to worry about the inflection points-- but it is true that there are four)

Answer 2

C, if u graph it u can see

Answer 3

c

Answer 4

E. I think it' a parabola

Answer 5

F. WGAF.