Generalized_Mean_Value_Theorem

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Definition

If $f$ and $g$ are continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists a point $c \in (a,b)$ where

$$ [f(b) - f(a)]g'(c) = [g(b) - g(a)]f'(c) $$

If $g'$ is never zero on $(a,b)$, this can be written as

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

Proof

Define $h(x) = [f(b) - f(a)]g(x) - [g(b) - g(a)]f(x)$.

Then $h(a) = f(b)g(a) - f(a)g(a) - g(b)f(a) + g(a)f(a) = f(b)g(a) - g(b)f(a)$

And $h(b) = f(b)g(b) - f(a)g(b) - g(b)f(b) + g(a)f(b) = g(a)f(b) - f(a)g(b) = h(a)$

By Rolle’s Theorem, there exists $c \in (a,b)$ with $h'(c) = 0$, giving the result.