Rolle's Theorem in Calculus: Finding Points with Zero Slope

Learn about Rolle's Theorem in calculus, a special case of the Mean Value Theorem. This guide explains the conditions for Rolle's Theorem, provides a clear explanation and proof, and illustrates its application with examples.

Rolle's Theorem in Discrete Mathematics

Statement of Rolle's Theorem

Rolle's Theorem is a special case of the Mean Value Theorem. It states: If a function f(x) is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point c in the open interval (a, b) such that f'(c) = 0.

In simpler terms: If a smooth curve starts and ends at the same height, there must be at least one point somewhere in between where the tangent to the curve is perfectly horizontal (slope = 0).

Proof of Rolle's Theorem

We consider two cases:

1. Case 1: f(x) = 0 for all x in [a, b]: If the function is a horizontal line, then its derivative is 0 everywhere, and any point c in (a,b) satisfies the theorem.
2. Case 2: f(x) ≠ 0 for some x in (a, b): Because f(x) is continuous on the closed interval [a, b], it must attain both its maximum and minimum values on this interval (Extreme Value Theorem). Since f(a) = f(b) = 0 and f(x) is not identically zero, the maximum or minimum value (or both) must occur at some point c within the open interval (a, b). At this point c, the function has a local extremum. Because f(x) is differentiable at c, Fermat's theorem tells us that f'(c) = 0.

Example 1: Verifying Rolle's Theorem

(This example, verifying Rolle's theorem for the function y = x² + 3 on the interval [-2, 2], is given in the original text and should be included here. The solution showing that the function satisfies the conditions of Rolle's theorem and finding the point c where f'(c) = 0 should be clearly shown.)

Example 2: Finding an Unknown Value

(This example, finding the value of n given that Rolle's Theorem applies to y = mx² + nx + k, with k = 0 and the Rolle's point c = 0, is given in the original text and should be included here. The solution should be provided.)

Geometric and Algebraic Interpretations

Geometric Interpretation: If a curve is continuous between two points with equal y-values, there's at least one point where the tangent line is horizontal.

Algebraic Interpretation: If f(x) is a polynomial and f(a) = f(b) = 0, then there's at least one root of f'(x) = 0 between a and b.

Examples: Applying Rolle's Theorem

Example 1: Verifying Rolle's Theorem

(This example, verifying Rolle's theorem for the function y = x² + 5 on the interval [-3, 3], is given in the original text and should be included here. The solution showing that the function satisfies the conditions of Rolle's theorem and finding the point c where f'(c) = 0 should be clearly shown.)

Example 2: Finding an Unknown Value

(This example, finding the value of j given that Rolle's Theorem applies to y = ix² + jx + k, with k = 0 and the Rolle's point c = 0, is given in the original text and should be included here. The solution should be provided.)

Conclusion

Rolle's Theorem is a fundamental result in calculus that guarantees the existence of a point with a horizontal tangent under specific conditions. It serves as a stepping stone to understanding the more general Mean Value Theorem.