# Changes

## Lagrange mean value theorem

, 00:29, 4 September 2011
Created page with "==Statement== Suppose $f$ is a function defined on a closed interval $[a,b]$ (with $a < b$) such that the following two conditions hold: # ..."
==Statement==

Suppose $f$ is a function defined on a [[closed interval]] $[a,b]$ (with $a < b$) such that the following two conditions hold:

# $f$ is a [[continuous function]] on the [[closed interval]] $[a,b]$ (i.e., it is right continuous at $a$, left continuous at $b$, and two-sided continuous at all points in the open interval $(a,b)$).
# $f$ is a [[differentiable function]] on the [[open interval]] $(a,b)$, i.e., the derivative exists at all points in $(a,b)$. Note that we ''do not'' require the [[derivative]] of $f$ to be a continuous function.

Then, there exists $c$ in the open interval $(a,b)$ such that the derivative of $f$ at $c$ equals the [[difference quotient]] $\Delta f(a,b)$. More explicitly:

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

Geometrically, this is equivalent to stating that the [[tangent line]] to the [[graph]] of $f$ at $c$ is parallel to the [[chord]] joining the points $(a,f(a))$ and $(b,f(b))$.

Note that the theorem simply guarantees the existence of $c$, and does not give a formula for finding such a $c$ (which may or may not be unique).

==Related facts==

* [[Rolle's theorem]]
* [[Zero derivative implies locally constant]]
* [[Fundamental theorem of calculus]]
* [[Positive derivative implies increasing]]
* [[Increasing and differentiable implies nonnegative derivative]]
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