# Difference between revisions of "Lagrange mean value theorem"

From Calculus

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* [[Positive derivative implies increasing]] | * [[Positive derivative implies increasing]] | ||

* [[Increasing and differentiable implies nonnegative derivative]] | * [[Increasing and differentiable implies nonnegative derivative]] | ||

+ | * [[Derivative of differentiable function satisfies intermediate value property]] |

## Revision as of 20:12, 7 September 2011

## Statement

Suppose is a function defined on a closed interval (with ) such that the following two conditions hold:

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

Then, there exists in the open interval such that the derivative of at equals the difference quotient . More explicitly:

Geometrically, this is equivalent to stating that the tangent line to the graph of at is parallel to the chord joining the points and .

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