The points of inflection occur where the second derivative is zero.
First find the first derivative.
#f (x) = x^3 + 3 x^2 - (27/x^2)#
#f (x) = x^3 + 3 x^2 - 27 (x^{-2})#
# {d f (x)} / {dx} = 3 x^2 + 3 * 2 x - 27*(-2)(x^{-3})#
# {d f (x)} / {dx} = 3 x^2 + 6 x + 54 x^{-3}#
or # {d f (x)} / {dx} = 3 x^2 + 6 x + (54 / {x^{-3}})#
Now the second.
# {d^2 f (x)} / {dx^2} = 3*2 x^1 + 6*1*x^0 +54*(-3) (x^{-4})#
# {d^2 f (x)} / {dx^2} = 6x + 6 -162 x^{-4}#
set this equal to zero.
# 0 = 6x + 6 -162 x^{-4}#
Multiply both sides by #x^4# ( allowed as long as #x != 0# and since the function blows up at zero, this is fine).
# 0 = 6x^5 + 6 x^4 -162#
Divide through by 6!
# 0 = x^5 + x^4 - 27# Go to a equation solver (like Maple, Mathcad or Matlab ) and find the 0's.
Check these (probably five) values in the function and the derivative to make sure they aren't doing anything foolish.
