For the function, use the First Derivative Test to determine if each critical point is a minimum, a maximum, or neither.
f (x) = -x3 + 3x2 - 3x
The derivative is
This is only zero when x = 1, and never undefined, so x = 1 is the only critical point. Since the quantity ( x - 1 )2 is positive for all x ≠ 1, the derivative
f'(x) = -3( x - 1 )2
is negative for all x ≠ 1. We find this numberline:
Since f is decreasing, levels out, and then decreases again, we have neither a minimum nor a maximum at x = 1.