Lagrange mean value theorem
From Calculus
Contents
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).
Related facts
- Rolle's theorem
- Zero derivative implies locally constant
- Fundamental theorem of calculus
- Positive derivative implies increasing
- Increasing and differentiable implies nonnegative derivative
- Derivative of differentiable function on interval satisfies intermediate value property
Facts used
- Continuous functions form a vector space
- Differentiable functions form a vector space
- Rolle's theorem
- Differentiation is linear
Proof
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Consider the function ![]() Then, ![]() ![]() ![]() |
Just plug in and check. Secretly, we obtained ![]() ![]() ![]() | |||
2 | Define ![]() ![]() ![]() |
||||
3 | ![]() ![]() |
Fact (1) | ![]() ![]() |
Steps (1), (2) | [SHOW MORE] |
4 | ![]() ![]() |
Fact (2) | ![]() ![]() |
Steps (1), (2) | [SHOW MORE] |
5 | ![]() |
Steps (1), (2) | [SHOW MORE] | ||
6 | There exists ![]() ![]() |
Fact (3) | Steps (3), (4), (5) | [SHOW MORE] | |
7 | For the ![]() ![]() |
Fact (4) | Steps (2), (6) | [SHOW MORE] | |
8 | ![]() ![]() ![]() |
Step (1) | Differentiate the expression for ![]() | ||
9 | ![]() |
Steps (7), (8) | Step-combination direct |