Example 3:

Let

, a = -√2, b = √2.

Then

f(a) = f(b) = 0.

Rolle's Theorem says that there is at least one point c in (-√2, √2) at which f'(c) = 0. As a matter of fact there are three such points,

c = -1, c = 0, c = 1.

We can find these points as follows:

f'(x) = 2x³ - 2x = 2x(x² - 1),

f'(x) = 0 when x = 0 or x = ±1.

The function is drawn in Figure 3.8.13.

Figure 3.8.13