How do use the first derivative test to determine the local extrema #f(x)=x^3 - 9x^2 + 27x#?

1 Answer
Jul 27, 2015

Answer:

This function has no local extrema.

Explanation:

#f(x) = x^3-9x^2+27x#

Has derivative:

#f'(x) = 3x^2 -18x+27 = 3(x^2-6x+9) = 3(x-3)^2#

Every local extremum occurs at a critical number.
(A number, #c#, in the domain of #f# at which either #f'(c) = 0# or #f'(c)# does not exist).

The only critical number is #3#, and the derivative does not change sign at #3# (this derivative is always non-negative).

So the function has no local extrema.