state lagrange mean value theorem and give its geometrical interpretation.is the therorem applicable to f(x)=|x-3|+2 in [2,3]
Answers
Answer:
The Mean Value Theorem (MVT)
Lagrange’s mean value theorem (MVT) states that if a function
f
(
x
)
is continuous on a closed interval
[
a
,
b
]
and differentiable on the open interval
(
a
,
b
)
,
then there is at least one point
x
=
c
on this interval, such that
f
(
b
)
−
f
(
a
)
=
f
′
(
c
)
(
b
−
a
)
.
This theorem (also known as First Mean Value Theorem) allows to express the increment of a function on an interval through the value of the derivative at an intermediate point of the segment.
Explanation:
Consider the auxiliary function
F
(
x
)
=
f
(
x
)
+
λ
x
.
We choose a number
λ
such that the condition
F
(
a
)
=
F
(
b
)
is satisfied. Then
f
(
a
)
+
λ
a
=
f
(
b
)
+
λ
b
,
⇒
f
(
b
)
−
f
(
a
)
=
λ
(
a
−
b
)
,
⇒
λ
=
−
f
(
b
)
−
f
(
a
)
b
−
a
.
As a result, we have
F
(
x
)
=
f
(
x
)
−
f
(
b
)
−
f
(
a
)
b
−
a
x
.
The function
F
(
x
)
is continuous on the closed interval
[
a
,
b
]
,
differentiable on the open interval
(
a
,
b
)
and takes equal values at the endpoints of the interval. Therefore, it satisfies all the conditions of Rolle’s theorem. Then there is a point
c
in the interval
(
a
,
b
)
such that
F
′
(
c
)
=
0.
It follows that
f
′
(
c
)
−
f
(
b
)
−
f
(
a
)
b
−
a
=
0
or
f
(
b
)
−
f
(
a
)
=
f
′
(
c
)
(
b
−
a
)
.