Rolle's Theorem and cubic: Apply Rolle's Theorem to the cubic shown:

.

Since f is differentiable everywhere, Rolle's Theorem says that f '(c)= 0 for a value of x in the interval (-2,1) and also for a point in the interval (1,3). So we need to find the first derivative and set it to 0:

or at either value of c: .

Using the quadratic formula, we get

or 2.12

Test Problem: Apply Rolle's Theorem to

a) -1.00, -3.00
b) -2.00, 2.50
c) -2.08, 2.08
d) -1.00, 4.00