What are the points of inflection, if any, of #f(x) = 5x^3 + 30x^2 - 432x #?

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Answer 1 · 2018-05-22T01:35:03

#(-2,944)#

points of inflection are where #f''(x)# changes signs

in this problem, #f'(x)=15x^2+60x-432#, #f''(x)=30x+60#

since #f''(x)# is linear, you can find where it changes signs by setting it equal to 0.

#f''(x)=30x+60=0#
#x=-2#

if you plug in x-values near -2, you can see that #f''(x)# changes signs at x=-2:

#f''(-1.9)=3>0#
#f''(-1.99)=0.3>0#

#f''(-2.1)=-3<0#
#f''(-2.01)=-0.3<0#

to find point of inflection: #f(-2)=944#
the point of inflection is #(-2,944)#

check with the graph of #f(x)#:
graph{5x^3+30x^2-432x [-14, 10, -10000, 10000]}

it seems to change concavity around #x=-2#