Lagrange mean value theorem

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Statement

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

  1. 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 ).
  2. 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).

Related facts

Facts used

  1. Continuous functions form a vector space
  2. Differentiable functions form a vector space
  3. Rolle's theorem
  4. Differentiation is linear

Proof

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 Consider the function
.
Then, is a linear (and hence a continuous and differentiable) function with and
Just plug in and check. Secretly, we obtained by trying to write the equation of the line joining the points and .
2 Define on , i.e., .
3 is continuous on Fact (1) is continuous on Steps (1), (2) [SHOW MORE]
4 is differentiable on Fact (2) is differentiable on Steps (1), (2) [SHOW MORE]
5 Steps (1), (2) [SHOW MORE]
6 There exists such that . Fact (3) Steps (3), (4), (5) [SHOW MORE]
7 For the obtained in step (6), Fact (4) Steps (2), (6) [SHOW MORE]
8 for all . In particular, . Step (1) Differentiate the expression for from Step (1).
9 Steps (7), (8) Step-combination direct