Mean value theorem establishes the existence of a point, in between two points, at which the tangent to the curve is parallel to the secant joining those two points of the curve.

**Mean Value Theorem**

Mean
value theorem establishes the existence of a point, in between two points, at
which the tangent to the curve is parallel to the secant joining those two
points of the curve. We start this section with the statement of the
intermediate value theorem as follows :

** **

**Theorem 7.1
(Intermediate value theorem)**

If *f* is continuous on
a closed interval [*a* , *b*] , and *c* is any number between *f*
(*a*) and *f* (*b*) inclusive, then
there is at least one number *x* in the
closed interval [*a* , *b*] , such that *f* ( *x*) = *c* .

**Theorem 7.2 (Rolleâ€™s
Theorem)**

Let *f* ( *x*) be continuous on a closed interval [*a* , *b*]
and differentiable on the open interval (*a*
, *b*)

If *f* (*a*) = *f* (*b*) , then there is at least one point *c* âˆˆ (*a*, *b*) where *f*â€™(*c*)
=0.

Geometrically
this means that if the tangent is moving along the curve starting at* x *= a towards as in Figâ€‰7.2 *x* =*
b *then there exists a c âˆˆ (
*a, b*) at which the tangent is parallel
to the* x *-axis.

**Theorem 7.3**

Let *f (x)* be
continuous in a closed interval [a, b] and differentiable in the open interval (a , b) (where* f *(a),* f *(b) are not necessarily equal). Then there exist at least one
point c âˆˆ*( a , b)* such that,* *

* f* â€²(c) = *f *(b) âˆ’* f *(a) / b âˆ’ a ... (6)

** **

**Remark**

If *f *(*a*)* *=* f *(*b*)
then Lagrangeâ€™s Mean Value Theorem gives the Rolleâ€™s theorem. It is also known
as **rotated
Rolleâ€™s Theorem**.

**Remark**

A
physical meaning of the above theorem is the number *f* (b) âˆ’* f *(a) / b âˆ’ a = can
be thought of as the average rate of
change in *f* ( *x*) over (*a*, *b*) and *f* (*c*) as an instantaneous
change.

A
geometrical meaning of the Lagrangeâ€™s mean value theorem is that the
instantaneous rate of change at some interior point is equal to the average
rate of change over the entire interval. This is illustrated as follows :

If a car
accelerating from zero takes just 8 seconds to travel 200 m, its average
velocity for the 8 second interval is 200/8 = 25 m/s. The Mean Value Theorem says
that at some point during the travel the speedometer must read exactly 90 km/h
which is equal to 25 m/s.

** **

**Theorem 7.4**

If *f* ( *x*) is continuous in closed interval [*a* , *b*]
and differentiable in open interval (*a*
, *b*) and if *f *â€²(* x*)* *>* *0,* *âˆ€*x *âˆˆ* *(*a*,* b*)* *, then for,* x*_{1}* *,*
x*_{2}* *âˆˆ[*a *,* b*]*
*, such that* x*_{1}* *<* x*_{2}* *we have,* f *(*x*_{1}* *)* *<* f *(*x*_{2}* *)* *.

**Proof**

By the
mean value theorem, there exists a *c* âˆˆ ( *x*_{1} , *x*_{2}
) âŠ‚ (*a*, *b*) such that,

* f *(*x*_{2}* *)* *âˆ’* f *(*x*_{1})
**/**
_{ }*x*_{2}* *âˆ’* x*_{1 }= *f* â€²(*c*)_{}

Since *f* â€²(*c*)
>
0 , and *x*_{2} âˆ’
*x*_{1} >
0 we have *fâ€‰*(*x*_{2}) â€“ *fâ€‰*(*x*_{1}) > 0.

We
conclude that, whenever *x*_{1}
<
*x*_{2} , we have *f* (*x*_{1}
) <
*f* (*x*_{2} ) .

**Remark**

If *f* â€²( *x*) < 0, âˆ€*x* âˆˆ (*a*, *b*) , then for, *x*_{1} , *x*_{2} âˆˆ [*a*,
*b*] , such that *x*_{1} < *x*_{2}
we have, *f* (*x*_{1} ) > *f*
(*x*_{2} ) .

The
proof is similar.

** **

Tags : Differential Calculus | Mathematics , 12th Maths : UNIT 7 : Applications of Differential Calculus

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12th Maths : UNIT 7 : Applications of Differential Calculus : Mean Value Theorem | Differential Calculus | Mathematics

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