f'(x) = 12x³ − 48x² + 48x = 12x(x² − 4x + 4) = 12x(x−2)². The factor (x−2)² is never negative, so the sign of f'(x) is controlled entirely by x: negative for x<0, positive for x>0. At x=0, f' changes sign (from − to +), so x=0 IS a genuine extremum (a local minimum). At x=2, f'(x)=0 but doesn't change sign (it's a repeated root, staying non-negative on both sides), so x=2 is a stationary inflection point, not an extremum. Hence there is exactly 1 distinct extremum.