Q: State And Prove Rolles Theorem
Rolle's Theorem states that if a function \(f\) 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 interval \((a, b)\) such that \(f'(c) = 0\).
To prove Rolle's Theorem, we can use the Mean Value Theorem. By the Mean Value Theorem, since \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\), there exists a point \(c\) in \((a, b)\) where the derivative of \(f\) at that point is equal to the average rate of change of \(f\) over the interval \([a, b]\). Mathematically, this can be expressed as:
\[f'(c) = \frac{f(b) - f(a)}{b - a}\]
Since \(f(a) = f(b)\), the numerator of the right-hand side becomes 0. Thus, we have:
\[f'(c) = 0\]
Therefore, there exists at least one point \(c\) in the interval \((a, b)\) such that \(f'(c) = 0\), proving Rolle's Theorem.