Cauchy's Mean Value Theorem - Notes and Analytical Proof

Rolle's Mean Value Theorem (RMVT)

• Statement and Hypotheses:
  • Let $f(x)$ be a real-valued function defined on a closed interval $[a, b]$.
  • Hypothesis 1: $f(x)$ is continuous on the closed interval $[a, b]$.
  • Hypothesis 2: $f(x)$ is differentiable on the open interval $(a, b)$.
  • Hypothesis 3: The endpoint function values are equal, such that $f(a) = f(b)$.
• Conclusion:
  • If all three hypotheses are satisfied, there exists at least one point $c \times (a, b)$ such that the first derivative vanishes:     $f'(c) = 0$

Lagrange's Mean Value Theorem (LMVT)

• Statement and Hypotheses:
  • Let $f(x)$ be a real-valued function defined on a closed interval $[a, b]$.
  • Hypothesis 1: $f(x)$ is continuous on the closed interval $[a, b]$.
  • Hypothesis 2: $f(x)$ is differentiable on the open interval $(a, b)$.
• Conclusion:
  • There exists at least one point $c \in (a, b)$ such that the derivative at $c$ equals the average rate of change over the interval $[a, b]$:     $f'(c) = \frac{f(b) - f(a)}{b - a}$
• Relationship to Rolle's Theorem:
  • Lagrange's Mean Value Theorem is a generalization of Rolle's Mean Value Theorem obtained by relaxing the restriction that $f(a) = f(b)$.

Cauchy's Mean Value Theorem (CMVT)

• Overview:
  • Cauchy's Mean Value Theorem, also known as the Extended Mean Value Theorem or Generalized Mean Value Theorem, generalizes Lagrange's Mean Value Theorem to a pair of functions $f(x)$ and $g(x)$.
• Hypotheses:
  • Let $f(x)$ and $g(x)$ be two real-valued functions defined on a closed interval $[a, b]$.
  • Hypothesis 1: Both $f(x)$ and $g(x)$ are continuous on the closed interval $[a, b]$.
  • Hypothesis 2: Both $f(x)$ and $g(x)$ are differentiable on the open interval $(a, b)$.
  • Hypothesis 3: The derivative of the denominator function satisfies $g'(x) \neq 0$ for all $x \in (a, b)$.
• Conclusion:
  • There exists at least one interior point $c \in (a, b)$ such that:     $\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}$
• Key Properties and Non-zero Denominator Guarantee:
  • If $g(b) = g(a)$, then by Rolle's Theorem applied to $g(x)$, there would exist some point where $g'(c) = 0$. Because Hypothesis 3 explicitly forbids $g'(x) = 0$ anywhere in $(a, b)$, it guarantees that $g(b) \neq g(a)$, ensuring the denominator $g(b) - g(a)$ is strictly non-zero.

Analytical Proof of Cauchy's Mean Value Theorem

• Construction of the Auxiliary Function:
  • Define an auxiliary function $F(x)$ on the interval $[a, b]$ as a linear combination of $f(x)$ and $g(x)$:     $F(x) = f(x) + A \cdot g(x)$
  • Here, $A$ is a constant parameter chosen specifically so that $F(x)$ satisfies the boundary condition $F(a) = F(b)$.
• Evaluation of Constant $A$:
  • Evaluate $F(x)$ at the endpoints $x = a$ and $x = b$:     $F(a) = f(a) + A \cdot g(a)$$F(b) = f(b) + A \cdot g(b)$
  • Equate $F(a)$ and $F(b)$:     $f(a) + A \cdot g(a) = f(b) + A \cdot g(b)$
  • Rearrange the terms to isolate $A$:     $f(b) - f(a) = -A \cdot (g(b) - g(a))$$A = -\frac{f(b) - f(a)}{g(b) - g(a)}$
• Verification of Rolle's Theorem Hypotheses for $F(x)$:
  • Continuity: Since $f(x)$ and $g(x)$ are continuous on $[a, b]$, $F(x) = f(x) + A \cdot g(x)$ is continuous on $[a, b]$.
  • Differentiability: Since $f(x)$ and $g(x)$ are differentiable on $(a, b)$, $F(x)$ is differentiable on $(a, b)$.
  • Equal Boundary Values: $F(a) = F(b)$ holds by the definition of $A$
• Application of Rolle's Theorem:
  • Because $F(x)$ satisfies all conditions of Rolle's Mean Value Theorem on $[a, b]$, there exists at least one point $c \in (a, b)$ such that:     $F'(c) = 0$
• Derivative Calculation and Final Substitution:
  • Differentiating $F(x)$ with respect to $x$:     $F'(x) = f'(x) + A \cdot g'(x)$
  • Evaluating at $x = c$:     $F'(c) = f'(c) + A \cdot g'(c) = 0$$f'(c) = -A \cdot g'(c)$
  • Substituting $A = -\frac{f(b) - f(a)}{g(b) - g(a)}$ into the derivative expression:     $f'(c) = -\left(-\frac{f(b) - f(a)}{g(b) - g(a)}\right) \cdot g'(c)$$f'(c) = \frac{f(b) - f(a)}{g(b) - g(a)} \cdot g'(c)$
  • Dividing both sides by $g'(c)$ (since $g'(c) \neq 0$):     $\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}$

Geometric Interpretation of Cauchy's Mean Value Theorem

• Parametric Curve Representation:
  • Consider a curve defined parametrically by $x = g(t)$ and $y = f(t)$ for $t \in [a, b]$.
  • The endpoints of the curve corresponding to parameter values $t = a$ and $t = b$ are $(g(a), f(a))$ and $(g(b), f(b))$.
• Secant Line Slope:
  • The slope of the secant line (chord) connecting the two endpoints $(g(a), f(a))$ and $(g(b), f(b))$ is given by:     $\text{Slope of Secant} = \frac{f(b) - f(a)}{g(b) - g(a)}$
• Tangent Line Slope:
  • By parametric differentiation, the slope of the tangent line to the curve at any parameter value $t$ is given by:     $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{f'(t)}{g'(t)}$
• Geometric Meaning:
  • Cauchy's Mean Value Theorem guarantees the existence of at least one point $t = c \in (a, b)$ where the tangent line to the parametric curve is parallel to the secant line connecting the endpoints of the curve.
• Reduction to Lagrange's Mean Value Theorem:
  • If $g(x) = x$, then $g'(x) = 1$, $g(a) = a$, and $g(b) = b$
  • Substituting $g(x) = x$ into Cauchy's formula yields:     $\frac{f'(c)}{1} = \frac{f(b) - f(a)}{b - a}$$f'(c) = \frac{f(b) - f(a)}{b - a}$
  • This confirms that Lagrange's Mean Value Theorem is a direct special case of Cauchy's Mean Value Theorem.