Lagrange's Mean Value Theorem

Lagrange's mean value theorem or LMVT is considered one of the most important results in real analysis. An elegant proof of the Fundamental Theorem of Calculus can be given using LMVT.

Statement

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $f$ be continous on $[a,b]$ and differentiable on $(a,b)$ .

Then $\exists$ $c\in (a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$

Proof

We reduce the problem to the Rolle's theorem by using an 'auxillary function'.

Consider $g(x)=f(x)-\frac{f(b)-f(a)}{b-a}(x-a)$

note that $g(a)=g(b)=f(a)$

By Rolle's theorem, $\exists\; c\in (a,b)$ such that $g'(c)=0$

i.e. $f'(c)-\frac{f(b)-f(a)}{b-a}=0$

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

QED

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Lagrange's Mean Value Theorem

Statement

Proof

See Also