The roots of the polynomial x^3 − 2x^2 − 4x + 2 are:
x1 = 0.42801
x2 = −1.51414
x3 = 3.08613
x1 and x2 are in the desired interval [-2, 2]
f'(x) = 3x^2 - 4x - 4
so we have:
3x^2 - 4x - 4 = 0
x = ( 4 +- √(16 + 48) )/6
x_1 = -4/6 = -0.66
x_ 2 = 2
According to Rolle's theorem, we have one point in between:
x1 = 0.42801 and x2 = −1.51414
where f'(x) = 0, and that is
x_1 = -0.66so we see that Rolle's theorem holds in our function.
