By power rule, f'(x) = 12x3 - 12x = 12x (x2 - 1) = 12x (x + 1) (x - 1) .
Setting f'(x) = 0 gives x = -1 , 0 , and 1.
Because f'(x) is an odd degree polynomial with a + lead coefficient, it starts in QIII and ends in QI. Thus, it starts below the x-axis. Its three zeros are all single multiplicity, which means the graph crosses the x-axis at all three zeros.
This analysis leads us to conclude that f' goes from - to + at x = -1 and 1, which means f has local minima there (both are also the absolute minimum). f' goes from + to - at x = 0, which is f's only max.
These conclusions are also in keeping with the fact that f is an even function so its graph has y-axis symmetry.
