Rolle's Theorem and Its Proof

Study Notes

Statement of the Theorem

Rolle's Theorem states that if a function ( f(x) ) satisfies the following conditions:
1. ( f(x) ) is continuous on the closed interval ([a, b]).
2. ( f(x) ) is differentiable on the open interval ((a, b)).
3. ( f(a) = f(b) ).
Then, there exists at least one point ( c ) in the interval ((a, b)) such that ( f'(c) = 0 ).

Proof of Rolle's Theorem

Existence of Maximum and Minimum: By the Extreme Value Theorem, since ( f(x) ) is continuous on ([a, b]), it attains a maximum and a minimum.
Equality Condition: Since ( f(a) = f(b) ), the maximum and minimum must occur within the interval (i.e., at some point ( c ) where ( f(c) ) is either maximum or minimum, but ( f(c) = f(a) ) or ( f(c) = f(b) )).
Critical Points: If ( f(c) ) is the maximum or minimum and occurs at an interior point ( c ), then by the first derivative test, ( f'(c) = 0 ).
Conclusion: Thus, there exists at least one ( c ) in ((a, b)) where ( f'(c) = 0 ).

Geometric Interpretation

The graph of ( f(x) ) starts and ends at the same height (i.e., ( f(a) = f(b) )).
The derivative ( f'(c) = 0 ) indicates that at point ( c ), there is a horizontal tangent line.
Geometrically, this means there is at least one point in between ( a ) and ( b ) where the slope of the tangent is flat, representing a local maximum or minimum.

Applications in Calculus

Finding Critical Points: Used to locate points where the function's slope is zero, helping to identify local extrema.
Optimization Problems: Assists in solving problems related to finding maximum and minimum values of functions on closed intervals.
Intermediate Value Theorem: Provides a connection to other theorems in calculus and reinforces the idea of continuity and differentiability.
Analyzing Function Behavior: Important for understanding the characteristics and behavior of differentiable functions, aiding in sketching graphs and predicting function trends.

Statement of the Theorem

Rolle's Theorem applies to functions ( f(x) ) with specific criteria.
Conditions for Rolle's Theorem:
- ( f(x) ) must be continuous on the closed interval ([a, b]).
- ( f(x) ) must be differentiable on the open interval ((a, b)).
- Endpoints must have equal values: ( f(a) = f(b) ).
If conditions are met, there exists at least one point ( c ) within ((a, b)) such that the derivative ( f'(c) = 0 ).

Proof of Rolle's Theorem

Existence of Maximum and Minimum:
- The Extreme Value Theorem ensures that continuous functions achieve both maximum and minimum values on closed intervals.
Equality Condition:
- Given ( f(a) = f(b) ), the maximum or minimum value must occur at some internal point ( c ) within the interval.
Critical Points:
- If the maximum or minimum occurs at point ( c ), the first derivative test confirms that ( f'(c) = 0 ).
Conclusion:
- This leads to the conclusion that there is at least one point ( c ) in the interval ((a, b)) where the slope of the function is zero.

Geometric Interpretation

The function's graph rises and falls to the same height at points ( a ) and ( b ), indicating ( f(a) = f(b) ).
A horizontal tangent line at point ( c ) implies ( f'(c) = 0 ), indicating a flat slope.
This flatness represents either a local maximum or minimum occurring between points ( a ) and ( b ).

Applications in Calculus

Finding Critical Points:
- Useful in identifying points where the slope is zero, corresponding to local extremes.
Optimization Problems:
- Aids in determining maximum and minimum function values over specified intervals.
Intermediate Value Theorem Connection:
- Relates to the continuity and differentiability principles in calculus.
Analyzing Function Behavior:
- Essential for understanding differentiable function characteristics and for sketching graphs, aiding in predicting trends.

Description

Explore the fundamental concepts of Rolle's Theorem, which asserts the existence of at least one point where the derivative of a function equals zero, given certain conditions. This quiz focuses on understanding the conditions, proof, and implications of this theorem in calculus.