We need to check that *f* satisfies all the hypotheses of Rolle's Theorem. *f* is a polynomial, so *f* is continuous on [-2, 2].*f* is differentiable on (-2, 2), since we found that *f ' *(*x*) = 2*x*.*f*(-2) = 4 and *f*(2) = 4, so *f*(-2) = *f*(2).
Since *f* satisfies all the hypotheses of Rolle's Theorem, Rolle's Theorem says there must be some *c* in (-2, 2) for which *f *' (*c*) = 0. In this case, *f*(*x*) = *x*^{2} has a "turn-around point" at *x* = 0, so *f ' *(0) = 0. We can see this from looking at the graph or from finding *f ' *(0), but not from Rolle's Theorem. Rolle's Theorem doesn't tell us where *f '* is zero, just that it is somewhere. | |