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Concavity of this function? please help?
where is y=x/(x^3-1) concave up and where is it concave down. please show work if you can. I know you have to take the second derivative but i cannot figure out the right answer.
1 Answer
- NozarLv 68 years agoFavorite Answer
y '=[(x^3-1) -3x^2(x)] /(x^3-1)^2=
=y '=(-2x^3 -1)/(x^3-1)^2
y "=(-6x^2)(x^3-1)^2 -2(3x^2)(x^3-1)(-2x^3-1) /(x^3-1)^4
y "=(x^3-1)[(-6x^2(x^3-1)-6x^2(-2x^3-1)]/(x^3-1)^4
y "=(x^3-1)(-6x^5+6x^2+12x^5+6x^2) /(x^3-1)^4
y "=(6x^5+12x^2) /(x^3-1)^3 ,y "=0
6x^2(x^3+1)/(x^3-1)^3=0
x=-1, x=0 & x=1
please see the table
x_________ -1_________________0______________+1
x^3+1 MIN 0 PLUS PLUS PLUS
x^2 PLUS PLUS PLUS PLUS
x^3-1 MIN MIN MIN MIN
__________________________________________________________
y " + - - -
con up<-1 down>-1
God bless you.