Question

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

Answer 1

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.