Extended Mean Value Theorem — Definition, Formula & Examples

The Extended Mean Value Theorem (also called Cauchy's Mean Value Theorem) states that for two functions continuous on a closed interval and differentiable on its interior, there exists a point where the ratio of their derivatives equals the ratio of their overall changes. It generalizes the standard Mean Value Theorem by replacing the simple difference quotient with a ratio involving two functions.

Let $f$ and $g$ be continuous on $[a, b]$ and differentiable on $(a, b)$ , with $g'(x) \neq 0$ for all $x \in (a, b)$ . Then there exists at least one $c \in (a, b)$ such that $\dfrac{f'(c)}{g'(c)} = \dfrac{f(b) - f(a)}{g(b) - g(a)}$ .

Key Formula

\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}

Where:

$f, g$ = Functions continuous on [a, b] and differentiable on (a, b)
$a, b$ = Endpoints of the closed interval
$c$ = A point in (a, b) guaranteed by the theorem
$g'(x) \neq 0$ = Required condition on the open interval to avoid division by zero

How It Works

Think of the ordinary Mean Value Theorem as the special case where

g(x) = x

. The Extended version lets you compare the rates of change of two functions simultaneously. To apply it, verify that both functions are continuous on

[a,b]

and differentiable on

(a,b)

, and that

g'(x) \neq 0

on that open interval. Then the theorem guarantees a point

c

where the ratio

f'(c)/g'(c)

matches the slope ratio of the two functions' net changes. This result is the key lemma used to prove L'Hôpital's Rule.

Worked Example

Problem: Let f(x) = x³ and g(x) = x² on [1, 2]. Find a value c in (1, 2) satisfying the Extended Mean Value Theorem.

Compute the right-hand side: Calculate the ratio of the net changes of f and g over [1, 2].

\frac{f(2) - f(1)}{g(2) - g(1)} = \frac{8 - 1}{4 - 1} = \frac{7}{3}

Set up the derivative ratio: Find f'(x) and g'(x), then form the ratio f'(c)/g'(c).

\frac{f'(c)}{g'(c)} = \frac{3c^2}{2c} = \frac{3c}{2}

Solve for c: Set the derivative ratio equal to the net-change ratio and solve.

\frac{3c}{2} = \frac{7}{3} \implies c = \frac{14}{9} \approx 1.556

Answer:

c = \dfrac{14}{9} \approx 1.556

, which lies in the interval

(1, 2)

as guaranteed.

Why It Matters

The Extended Mean Value Theorem is the theoretical backbone of L'Hôpital's Rule, which you use constantly to evaluate indeterminate limits in calculus. It also appears in real analysis proofs involving Taylor's theorem and error bounds for approximations.

Common Mistakes

Mistake: Forgetting to verify that g'(x) ≠ 0 on the open interval before applying the theorem.

Correction: If g'(x) = 0 somewhere in (a, b), the hypothesis fails and the conclusion is not guaranteed. Always check this condition, just as you check differentiability for the standard MVT.