Q: State And Prove the Rolle's Theorem With Diagram?

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State And Prove the Rolle's Theorem With Diagram? Also Explain Steps

Answer 1

Here is the statement of Rolle's Theorem: **Rolle's Theorem:** Let \(f(x)\) be a continuous function on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\). If \(f(a) = f(b)\), then there exists at least one \(c\) in the interval \((a, b)\) such that \(f'(c) = 0\). Now, let's prove Rolle's Theorem: **Proof:** 1. Since \(f(x)\) is a continuous function on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), it satisfies the conditions necessary to apply the Mean Value Theorem (MVT). 2. By the MVT, there exists a point \(c\) in the interval \((a, b)\) such that \(\frac{{f(b) - f(a)}}{{b - a}} = f'(c)\). 3. However, since \(f(a) = f(b)\), the numerator of the fraction on the left-hand side of the equation becomes zero, i.e., \(f(b) - f(a) = 0\). 4. Therefore, the equation simplifies to \(\frac{0}{{b - a}} = f'(c)\), which implies that \(f'(c) = 0\). 5. Hence, we have proven that if \(f(a) = f(b)\) for a function \(f(x)\) that is continuous on \([a, b]\) and differentiable on \((a, b)\), there exists at least one \(c\) in \((a, b)\) such that \(f'(c) = 0\), satisfying the conditions of Rolle's Theorem. Now, let's visualize this theorem with the help of a diagram: ``` | * | / f(x) | / | / | / -------------|---------------- a c b ``` In the diagram, the function \(f(x)\) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\). The points \(a\) and \(b\) have the same function values, i.e., \(f(a) = f(b)\). According to Rolle's Theorem, there exists at least one point \(c\) in the interval \((a, b)\) where the derivative \(f'(c) = 0\). This means that the slope of the tangent line to the curve at \(c\) is zero. The visual representation of the theorem shows that if the function starts and ends at the same height (same function values at \(a\) and \(b\)), there must be at least one point \(c\) within the interval where the tangent line is horizontal (slope \(f'(c) = 0\)).