Rolle's Theorem

What is Rolle's Theorem?

Rolle's Theorem is a fundamental result in differential calculus. It states that if a function satisfies certain conditions, then there exists at least one point in the interval where the derivative of the function is zero. Mathematically, if:

1. The function $f(x)$ is continuous on a closed interval $[a, b]$.
2. The function $f(x)$ is differentiable on the open interval $(a, b)$.
3. The values of the function at the endpoints of the interval are equal: $f(a) = f(b)$.

Then, there exists at least one point $c \in (a, b)$ such that:

Geometric Interpretation

Geometrically, Rolle's Theorem implies that if a function starts and ends at the same value on a given interval, then the curve has at least one point where the tangent is horizontal (slope = 0).

Conditions for Rolle's Theorem

The theorem applies only when the following conditions are satisfied:

• The function is continuous on $[a, b]$.
• The function is differentiable on $(a, b)$.
• The function satisfies $f(a) = f(b)$.

If any of these conditions are not met, Rolle's Theorem does not guarantee a point where $f'(c) = 0$.

Example Problems

Example 1: Polynomial Function

Verify Rolle's Theorem for $f(x) = x^2 - 4x + 3$ on the interval $[1, 3]$.

Solution:

Step 1: Check the conditions:

• Continuity: The function $f(x) = x^2 - 4x + 3$ is a polynomial, which is continuous everywhere.
• Differentiability: Polynomials are differentiable everywhere.
• Equal Endpoints: Calculate $f(1)$ and $f(3)$:
  $$f(1) = 1^2 - 4(1) + 3 = 0$$
  $$f(3) = 3^2 - 4(3) + 3 = 0$$
  Since $f(1) = f(3)$, the conditions are satisfied.

Step 2: Differentiate $f(x)$:

Step 3: Solve $f'(x) = 0$ to find $c$:

Since $x = 2 \in (1, 3)$, Rolle's Theorem is verified, and $c = 2$.

Example 2: Trigonometric Function

Verify Rolle's Theorem for $f(x) = \sin(x)$ on $[0, \pi]$.

Solution:

Step 1: Check the conditions:

• Continuity: $\sin(x)$ is continuous everywhere.
• Differentiability: $\sin(x)$ is differentiable everywhere.
• Equal Endpoints: Calculate $f(0)$ and $f(\pi)$:
  $$f(0) = \sin(0) = 0$$
  $$f(\pi) = \sin(\pi) = 0$$
  Since $f(0) = f(\pi)$, the conditions are satisfied.

Step 2: Differentiate $f(x)$:

Step 3: Solve $f'(x) = 0$:

Since $x = \frac{\pi}{2} \in (0, \pi)$, Rolle's Theorem is verified, and $c = \frac{\pi}{2}$.

Applications of Rolle's Theorem

• Proves the existence of roots in an equation.
• Forms the basis for the Mean Value Theorem.
• Helps in analyzing the behavior of curves.

Practice Problem

Problem:

Verify Rolle's Theorem for $f(x) = x^3 - 6x^2 + 11x - 6$ on $[1, 3]$.

Hint:

Follow the steps: check continuity, differentiability, and equal endpoints. Then find $f'(x)$ and solve $f'(x) = 0$.