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

Jul 27, 2015

This function has no local extrema.

Explanation:

$f \left(x\right) = {x}^{3} - 9 {x}^{2} + 27 x$

Has derivative:

$f ' \left(x\right) = 3 {x}^{2} - 18 x + 27 = 3 \left({x}^{2} - 6 x + 9\right) = 3 {\left(x - 3\right)}^{2}$

Every local extremum occurs at a critical number.
(A number, $c$, in the domain of $f$ at which either $f ' \left(c\right) = 0$ or $f ' \left(c\right)$ 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.