Rolle's Theorem states that 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 value \( c \) in \((a, b)\) such that \( f'(c) = 0 \).
To prove Rolle's Theorem, we will use the following steps:
1. Continuity: First, we need to show that the function \( f(x) \) is continuous on the closed interval \([a, b]\). Since this is given in the statement of the theorem, we don't need to prove it.
2. Differentiability: Next, we need to prove that the function \( f(x) \) is differentiable on the open interval \((a, b)\). This means that the derivative \( f'(x) \) exists for every value in \((a, b)\). We can use the differentiability of \( f(x) \) to prove this.
3. Equal function values: Since \( f(a) = f(b) \), it means that the function has the same value at the endpoints of the interval.
4. Application of Mean Value Theorem (MVT): By applying the MVT, we know that there exists at least one value \( c \) in \((a, b)\) where the derivative \( f'(c) \) is equal to the average rate of change of the function. The average rate of change is given by \( \frac{f(b) - f(a)}{b - a} = 0 \), since \( f(a) = f(b) \). Therefore, \( f'(c) = 0 \).
The diagram below illustrates the function \( f(x) \) on the interval \([a, b]\):
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a b
In the diagram, the function starts at a point on the left and ends at the same point on the right, with possible ups and downs in between. The point where \( f'(c) = 0 \) is represented by the point marked with an asterisk. This point is guaranteed to exist according to Rolle's Theorem.
Q: State And Prove Rolle's Theorem With Diagram?
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