Rolle’s Theorem

If a function is:

Continuous on [a, b].

Differentiable on (a, b).

f(a) = f(b).

Then, there exists a point, c (a, b) such that f'(c) = 0.

The graphical interpretation of Rolle's Theorem states that there is a point where the tangent is parallel to the x-axis.

Examples

1. Given the function , determine if Rolle's Theorem is varified on the interval [0, 3]?

First, verify that the function is continuous at x = 1.

Secondly, check if the function is differentiable at x = 1.

The function is not differentiable on the interval (0, 3) and therefore does not satisfy Rolle's Theorem.

2.Is Rolle's Theorem applicable to the function f(x) = ln (5 − x²) on the interval [−2, 2]?

First, calculate the domain of the function.

The function is continuous on the interval [−2, 2] and differentiable on (−2, 2), because the intervals are contained in .

Also, it is determined that f(−2) = f(2), therefore Rolle's theorem is applicable to this function.

3.Verify that the equation x⁷ + 3x + 3 = 0 has only one real solution.

The function f(x) = x⁷ + 3x + 3 is continuous and differentiable at ·

Intermediate Value Theorem.

f(−1) = −1

f(0) = 3

The equation has at least one solution in the interval (-1, 0).

Rolle's theorem.

f' (x) = 7x⁶ + 3

As the derivative is not annulled in any value, it contradicts Rolle's theorem and therefore only has one real root.