question about points of inflection derivatives (urgent)

y= 1/3 x^3 + x^2-15x +3

y'= x^2 +2x-15 =0

(x+5) (x-3)

x=-5 x=3

y'' = 2x +2 =0

check for max or min

2(-5)+2 =-8< 0 local max

2(3) +2= 8 >o local min

check for poi

y'''= 2 not= 0 so we have a poi check if its stationary

2x+2=0 2X=-2 x=- 2/2 x=-1

non stationary point of infecton

so if the was x=0 we have a stationary point of inflection?

because i the book is they have this

y=(x+4)^3

y'=3(x+4)^2 =o x=-4

y''= 6(x+4)=0 x=-4

y'''=6 is not = to 0 we have a poi

the order of the derivative is an odd number so at x=-4 we have a poi since y=0 when x=-1 the point is stationary

bit confused because i thought that when the 2nd derviative result is calculated and x=0 then it is stationary when its x not = 0 nonstationary.