What are the points of inflection, if any, of #f(x) =x^3 - 3x^2 + 3x#?
1 Answer
Explanation:
At a point of inflection the slope of the function must be equal to zero.
(Note that a slope of zero does not necessarily indicate a point of inflection; it could be a local minimum or local maximum).
If
then
so the only candidate as a possible point of inflection is
There are numerous ways to check if this point is a local minimum, point of inflection, or local maximum.
First and second derivative tests are often suggested.
Here is an alternative:
Suppose
#barx# is our candidate to be tested
Pick any value#v_l < barx# such that#v_l# is greater than any other value#x_l < barx# for which#f'(x_l)=0#
and
any value#v_r > barx# such that#v_r# is less than any other value#x_r > barx# for which#f'(x_r)=0#
#color(white)("XXX")# (this is simpler than it first sounds).For our example, since there are no values other than
#x=1# for which#f'(x)=0# we can pick any values#v_l < 1# and#v_r > 1# .
For this example, choosing
but
Therefore