Rolle's Theorem and Applications

What does Rolli's Theorem guarantee for a continuous and differentiable function on the interval [a,b], where f(a) = f(b)?

In the context of Rolli's Theorem, what does it mean if f’(c) = 0 at point c?

Which equation follows from applying Rolli's Theorem to the function f(x) = x^3 - 3x^2 + 2x + 2 on [0,1]?

What is indicated by the statement that there exists a number c ∈ (a,b) such that f’(c) = (f(b) - f(a)) / (b - a) in the Mean Value Theorem?

For the function f(x) = sin(x) in the interval [0, π], what is the derived value of c that satisfies the conclusion of Rolli's Theorem?

Rolle's Theorem Condition 1

Rolle's Theorem Condition 2

Mean Value Theorem Condition

Rolle's Theorem

Statement: If a function f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists a number c ∈ (a, b) such that f'(c) = 0.
Geometric Interpretation: The tangent to the graph of f at x = c is parallel to the x-axis. This means the graph has a horizontal tangent at x = c.

Examples and Applications

Example 1: Find a value of c satisfying the conclusion of Rolle's Theorem for f(x) = x³ - 3x² + 2x + 2 on [0, 1].
- f(0) = 2, f(1) = 2.
- f'(x) = 3x² - 6x + 2.
- Setting f'(c) = 0, we get 3c² - 6c + 2 = 0.
- Solving for c, c = (1 ± √3)/3
Example 2: Find a value of c satisfying the conclusion of Rolle's Theorem for f(x) = sin x in [0, π].
- f(0) = 0, f(π) = 0.
- f'(x) = cos x.
- Setting f'(c) = 0, we get cos c = 0.
- Solving for c, c = π/2.
Example 3: Find a value of c for f(x) = x³ - x + 1 on [0, 1].
- f(0) = 1, f(1) = 1
- f'(x) = 3x² - 1
- Setting f'(c) = 0, we get 3c² - 1 = 0.
- Solving for c , c = ± 1/√3

Mean Value Theorem

Statement: If a function f is continuous on [a, b] and differentiable on (a, b), then there exists a number c ∈ (a, b) such that:
f'(c) = (f(b) - f(a)) / (b - a)*.
Geometric Interpretation: There is a point on the graph of f between x = a and x = b where the tangent line is parallel to the secant line connecting the points (a, f(a)) and (b, f(b)).
Example 1: Find the value of c for f(x) = x³ - x² + x on [0, 2].
- f(0) = 0, f(2) = 4
- f'(x) = 3x² - 2x + 1
Setting f'(c) = (f(2) - f(0)) / (2 - 0), we get 3c² - 2c + 1 = 2.
Solving for c: 3c² - 2c - 1 = 0, c = (1 ± √2)/3.

Explore the fundamentals of Rolle's Theorem, covering its statement, geometric interpretation, and various applications. Solve examples to find values of c that satisfy the theorem for different functions. Perfect for those studying calculus concepts related to continuous and differentiable functions.