Problem 86 - Calculus Single Variable

Question

Cauchy's Mean Value Theorem states the following: If \(f(x)\) and \(g(x)\) are functions that are continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b),\) then there is a point \(\xi \in(a, b)\) such that \((g(b)-g(a)) f^{\prime}(\xi)=(f(b)-f(a)) g^{\prime}(\xi) .\) Notice that, if both \(g(a) \neq g(b)\) and \(g^{\prime}(\xi) \neq 0,\) then $$ \frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}=\frac{f(b)-f(a)}{g(b)-g(a)} $$ Complete the following outline to obtain a proof of Cauchy's Mean Value Theorem a. Let \(r(x)=\) $$ (f(b)-f(a))(g(x)-g(a))-(f(x)-f(a))(g(b)-f(a)) $$ Check that \(r(a)=r(b)\) b. Apply Rolle's Theorem to the function \(r\) on the interval \([a, b]\) to conclude that there is a point \(\xi \in(a, b)\) such $$ \text { that } r^{\prime}(\xi)=0 $$ c. Rewrite the conclusion of (b) to obtain the conclusion of Cauchy's Mean Value Theorem

Step-by-Step Solution

Answer

Verify \( r(a) = r(b) = 0 \), apply Rolle's theorem, differentiate \( r(x) \), and obtain \( \frac{f'(\xi)}{g'(\xi)} = \frac{f(b) - f(a)}{g(b) - g(a)} \).

Step 1: Define the Function r(x)

The exercise asks for defining the function \( r(x) \) as follows: \[ r(x) = (f(b) - f(a))(g(x) - g(a)) - (f(x) - f(a))(g(b) - g(a)) \] This function is constructed to leverage the properties of Cauchy's Mean Value Theorem.

Step 2: Verify Boundary Conditions

To verify \( r(a) = r(b) \):\ - Calculate \( r(a) \): \[(f(b) - f(a))(g(a) - g(a)) - (f(a) - f(a))(g(b) - g(a)) = 0\] - Calculate \( r(b) \): \[(f(b) - f(a))(g(b) - g(a)) - (f(b) - f(a))(g(b) - g(a)) = 0\] Since \( r(a) = 0 \) and \( r(b) = 0 \), it can be confirmed that \( r(a) = r(b) \).

Step 3: Apply Rolle's Theorem

Since \( r(a) = r(b) \), and \( r(x) \) is continuous on \([a, b]\) and differentiable on \((a, b)\), by Rolle's Theorem, there exists a point \( \xi \in (a, b) \) such that \( r'(\xi) = 0 \).

Step 4: Differentiate r(x) and Find r'(ξ)

Calculate \( r'(x) \) using product rule and chain rule:- \[ r'(x) = (f(b) - f(a))g'(x) - (f'(x))(g(b) - g(a)) \]Setting \( r'(\xi) = 0 \) gives:\[(f(b) - f(a))g'(\xi) = (f'(\xi))(g(b) - g(a))\]

Step 5: Conclude with Cauchy's Mean Value Theorem

Recognize that the equation from Step 4 is the conclusion of the Cauchy's Mean Value Theorem with:\[\frac{f'(\xi)}{g'(\xi)} = \frac{f(b) - f(a)}{g(b) - g(a)}\].This forms the final result needed to complete the theorem statement under the conditions that \( g(a) eq g(b) \) and \( g'(\xi) eq 0 \).

Key Concepts

Rolle's Theorem

Rolle's Theorem is a fundamental result in calculus that provides the necessary framework to understand certain properties of differentiable functions on a closed interval. This theorem states that if a function \( f \) is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and if it satisfies \( f(a) = f(b) \), then there exists at least one point \( \xi \) in \((a, b)\) such that \( f'((\xi) = 0 \). Rolle's Theorem is applied to prove Cauchy's Mean Value Theorem by showing that there is at least one point \( \xi \) where the derivative of a constructed function \( r(x) \) equals zero. This result is crucial, as it bridges the continuity and differentiability conditions towards proving equal rates of change at a certain \( \xi \).

Differentiability

Differentiability is a core concept in calculus that focuses on the ability of a function to have a derivative. A function is differentiable at a point if it has a defined derivative at that point.

• This means that the function must be smooth and not have any breaks, jumps, or sharp corners at the point itself.
• The concept of differentiability is essential for applying many calculus theorems, including Rolle's Theorem and, by extension, Cauchy's Mean Value Theorem.
• In the context of the theorem, since \( r(x) \) is differentiable on the interval \((a, b)\), it ensures that there is a smooth curve allowing a tangent line at any point \( \xi \) within the interval.

A differentiable function is automatically continuous, which is another condition needed for these theorems to hold. Differentiability intrinsically ensures that changes within the function are predictable and follow a defined trend.

Continuity

Continuity is a fundamental property in calculus that describes a function where its graph can be drawn without lifting the pen from the paper. Formally, a function \( f(x) \) is continuous at a point \( c \) if \(\lim_{x \to c} f(x) = f(c)\).
For Cauchy's Mean Value Theorem, continuity is required on the closed interval \([a, b]\). This ensures that no sudden jumps or breaks occur at any point within this interval.

• Continuity is imperative because it ensures that the behavior of the function is predictable as \( x \) moves from \( a \) to \( b \).
• Without this property, establishing a meaningful comparison of derivatives at a particular point \( \xi \) would be impossible.

Thus, ensuring continuity provides the initial groundwork needed to apply critical theorems like Rolle's Theorem and Cauchy's Mean Value Theorem.

Boundary Conditions

Boundary conditions describe the specific values that a function achieves at the endpoints of an interval. In the context of the Cauchy’s Mean Value Theorem, these conditions help to define and validate a function like \( r(x) \). To apply Rolle's Theorem effectively:

• It is critical to verify the boundary conditions \( r(a) = r(b) \).
• This verification ensures that the derivatives lead to meaningful interpretations.
• In exercises like this, solving for \( r(a) \) and \( r(b) \) will often result in zero, confirming that the conditions meet the \( f(a) = f(b) \) requirement needed for Rolle's Theorem.

When these boundary conditions are satisfied, it guarantees a valid application of further calculus theorems, allowing a deeper exploration into the function’s behavior within the interval.