What are the points of inflection, if any, of #f(x) =x^3 - 12x^2 #?

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Answer 1 · 2017-05-22T03:06:41

#x=4#

Points of inflection are where #f''(x)=0# or when #f''(x)# instantaneously changes from positive to negative or vice versa.

#f(x) = x^3-12x^2# is a polynomial -- it is a smooth curve so we don't have to worry about instantaneous jumps (e.g. asymptotes).

All we have to do is differentiate twice and set that equal to #0#.

#f'(x)=3x^2-24x#

#f''(x) = 6x-24#

Now set #f''(x)# equal to #0#.

#6x-24=0#
#6x=24#
#x=4#

This means that there is one point of inflection on the graph of #f(x)# and it is #x=4#.

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