3 The Mean Value Theorem That Will Change Your Life

Thus mean value theorem cannot be applied on this function in the interval [-2, 2]. An example where this version of the theorem applies is given by the real-valued cube root function mapping

{\displaystyle x\mapsto x^{1/3}}

, whose derivative tends to infinity at the origin. 34
Let

f
this link :
[
a
,
b
]

{\displaystyle f:[a,b]\to \mathbb {R} }

be a continuous function on the closed interval

[
a
,
b
]

{\displaystyle [a,b]}

, and differentiable on the open interval

(
a
,
b
)

{\displaystyle (a,b)}

, where

{\displaystyle ab}

. Since (f(b)−f(c))/(b−a) is the average change in the function over [a, b], and f'(c) is the instantaneous change at c, the mean value theorem states that at some interior point the instantaneous change is equal to average change of the function over the interval. Before we take a straight from the source at a couple of examples let’s think about a geometric interpretation of the Mean Value Theorem.
The above theorem implies the following:
Mean value inequality9For a continuous function

:
[
a
,
b
]

{\displaystyle {\textbf {f}}:[a,b]\to \mathbb {R} ^{k}}

, if

{\displaystyle {\textbf {f}}}

is differentiable on

(
a
,
b
)

{\displaystyle (a,b)}

, then
In fact, the above statement suffices for many applications and can be proved directly as follows.

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