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Introduction to Rolle’s Theorem

Rolle’s Theorem is a fundamental result in differential calculus that guarantees the existence of a point where the derivative of a function becomes zero, provided certain conditions are met.
This theorem serves as a cornerstone for the Mean Value Theorem and many other important results in analysis.

1. Statement of Rolle’s Theorem

Let f(x) be a function that satisfies the following conditions:

• It is continuous on the closed interval [a, b].
• It is differentiable on the open interval (a, b).
• f(a) = f(b).

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

f'(c) = 0

2. Geometric Interpretation

If the graph of a function starts and ends at the same height on an interval, the curve must have at least one point where the tangent line is horizontal.
This point corresponds to where the derivative equals zero.

3. Examples

Example 1: Quadratic Function

Function:

f(x) = x² - 4x + 3

On the interval [1, 3]:

f(1) = 0  
f(3) = 0

Thus, the conditions of Rolle’s Theorem are satisfied.

Derivative:

f'(x) = 2x - 4

Solving:

2x - 4 = 0 → x = 2

Therefore, c = 2 is the point guaranteed by Rolle’s Theorem.

Example 2: Trigonometric Function

Function:

f(x) = sin(x)

On the interval [0, π]:

f(0) = 0  
f(π) = 0

Derivative:

f'(x) = cos(x)

Solving:

cos(x) = 0 → x = π/2

4. Importance of Rolle’s Theorem

• Forms the basis of the Mean Value Theorem
• Helps analyze the behavior of functions
• Useful in proving uniqueness of roots
• Applied in solving differential equations
• Essential in mathematical analysis

5. Important Notes

• If any of the three conditions fail, the conclusion may not hold.
• The theorem guarantees at least one point, but there may be more.
• Rolle’s Theorem concerns the derivative, not the function’s values.

Conclusion

Rolle’s Theorem states that if a function meets the conditions of continuity, differentiability, and equal endpoint values, then there exists a point inside the interval where the derivative is zero.
This theorem is a foundational result in calculus and plays a crucial role in deeper mathematical analysis.