MATEMATIQKI VESNIK 51 (1999), 9{13

UDK 517.28 research paper

originalni nauqni rad

SOME MEAN VALUE THEOREMS IN TERMS OF AN INFINITESIMAL FUNCTION Branko J. Malesevic Abstract. In this paper we survey some mean value theorems in terms of an in nitesimal

function.

1. Introduction

Let f : [a; b] ! R be a function which is dierentiable in a segment [a; b] and dierentiable arbitrary number of times in a right neighbourhood of the point x = a. According to the following relation: y = f (x) ; f (a) = f 0 (a)(x ; a) + (x)(x ; a); where limx!a (x) = limx!a (x) = 0, let us introduce the function with: f x ;f a 0 x;a ; f (a); x 2 (a; b] (x) = (1) 0; x=a It is true that: f 0 (x) = f (x) ; f (a) ; f 0(a) = f (x) ; f (a) ; f (a)(x ; a) = R (a; x) (x 6= a); +

1

(

def

)

( )

1

x;a

1

2

x;a

x;a

where Rf (a; x) is the remainder of the second order in Taylor's formula. Then, 2

(2)

(x) ; (a) = lim f (x) ; f (a) ; f 0 (a)(x ; a) 0 (a) = xlim x!a !a x;a (x ; a) 0 0 f (x) ; f (a) 1 00 = xlim !a 2(x ; a) = 2 f (a): 1

1

1

2

We generalize this by induction as:

n (a) = n +1 1 f n (a); (

)

(

+1)

1

for an arbitrary n 2 N. The given formula is also a corollary of [3]. AMS Subject Classi cation : 26 A 24

9

(3)

10

B. J. Malesevic

Let us introduce analogously:

(x) =

def

2

1 x ;1 a x;a (

)

; 0 (a); x 2 (a; b]

( )

1

0;

(4)

x = a:

It is true that

(a) 0 (x) = (xx) ; ; a ; (a) f (x) ; f (a) ; f 0(a)(x ; a) ; f 00 (a)(x ; a) Rf (a; x) = (x ; a) (x 6= a); = (x ; a) 1

1

2

1

1

2

3

2

2

2

where Rf (a; x) is the remainder of the third order in Taylor's formula. According to the formula (3) the following formula for n derivative is true: n (a) = n +1 1 n (a) = (n + 1)(1 n + 2) f n (a): Next, let us introduce a sequence of functions: 3

th

(

)

(

2

+1)

(

k (x) =

def

k x ;k a x;a (

)

( )

0; For the function k let us assume: +1

; 0k (a); x 2 (a; b]

(k = 1; 2; . . . ):

x = a;

kn (a) = (n + 1)(n + 12) (n + k) f n k (a) (

+2)

1

)

(

(5)

(n 2 N)

+ )

and k (x) = Rkf (a; x)=(x ; a)k (x 6= a), where Rkf (a; x) is the remainder of (k +1) order in Taylor's formula. Then, according to the formula (3) and inductive assumption for k, we have: kn (a) = n +1 1 kn (a) = (n + 1)(n + 2)1 (n + k + 1) f n k (a) (6) and (x) = k (x) ; k (a) ; 0 (a) +1

+1

th

(

(

)

+1)

(

+ +1)

+1

k

+1

k x;a Rf (a; x) ; 0 ; k f k (a) (x ; a)k = k (x ; a)k

Rf (a; x) = (xk; a)k (x 6= a); (7) is the remainder of (k + 2) order in Taylor's formula. 1

+1

( +1)

+1

( +1)!

+2

+1

where Rkf

+1

th

+2

2. Mean value theorems Flett's theorem. We determine the conditions which are sucient for the existence of the point 2 (a; b) in which the function has an extreme value 1

11

Some mean value theorems in terms of an in nitesimal function

and the equality 0 ( ) = 0 is true. The re ection of the previous equality on the function f is Flett's equality: f 0 ( ) = f ( ) ; f (a) : (8) 1

;a

Namely, Flett [1] has proved that the condition: f 0 (a) = f 0 (b) (9) is sucient for the existence of the point 2 (a; b) so that (8) is true. Besides the previous condition, it is shown that the conclusion of the Flett's Theorem is true under the other conditions, as well. The following statement is true. Theorem. If one of the following two conditions is true: 0 (b) (b) < 0; T -condition; 1

1

(10)

1

and

0 (a) (b) < 0 M -condition; then there exists a point 2 (a; b), such that the following holds: ; f (a) : f 0 ( ) = f () ; a 1

1

(11)

1

(12)

If both of the conditions (10) and (11) are ful lled, then there exists at least two points i 2 (a; b), such that the following holds:

; f (a) f 0 (i ) = f (i ) ; a

(i = 1; 2):

i

(13)

From the condition of the corollary 3 of the theorem 2 [4]: 0 (a)[ (b) ; (a)] < 0; and 0 (b)[ (b) ; (a)] < 0; (14) we obtain the T and M conditions: 0 (a) (b) < 0; and 0 (b) (b) < 0; (15) respectively. According to the second part of the stated corollary for r = a the point 2 (a; b) exists, for which 0 ( ) = 0, i.e. f 0 ( ) = f ;;af a ; while for r = b the point 2 (a; b) exists, for which 0 ( ) = 0, i.e. f 0 ( ) = f ;;af a . If both of the stated conditions (10) and (11) are ful lled (T -M -condition), according to the rst part of the stated corollary, then two points i 2 (a; b) exist, for which 0 (i ) = 0, i.e. f 0 (i ) = f ;;fa a (i = 1; 2). We give extensions of the previous conditions on function n , with the suitable conclusions. Proof.

1

1

1

1

1

1

1

1

1

2

1

2

1

( 2)

2

1

( i)

i

( )

( )

2

1

( 1)

1

1

1

1

1

1

( )

12

B. J. Malesevic

Corollary 1 (Tn -condition). By applying T -condition on function n we get Tn -condition. Thus, the following implication is true: 0n (b) n (b) < 0 =) (9 2 (a; b)) 0n ( ) = 0: (16) The re ection of the previous relation on the original function f is given by implication: 1

nP ;1

n;j f (j) a (b ; a)j ; + f b n; f a ; f b ;f a Rf (a; b) < 0 n n j n b;a j nP ; ; (j) =) 9 2 (a; b) f ;;fa a = (1 ; nj ) f j a ( ; a)j; + f nn; f a j ( )

0

1

( )+(

0

1)

( )

( )

( )

+1

!

=2

( )

1

( )

( )

0

1

( )+(

0

1)

!

=2

(17)

:

( )

For n = 1; 2; 3 we get the corresponding implications: ; ; 0 f (a) ; f bb;;af a f 0 (b) ; f bb;;af a > 0 =) (9 2 (a; b)) f ;;af a = f 0 ( ); (18) ;f b f a f b ; f a f ; f a f f a f ; b;a R (a; b) < 0 =) (9 2 (a; b)) ;a = (19) and ;f a (b ; a) + f b f a ; f bb;;af a Rf (a; b) < 0 (20) =) (9 2 (a; b)) f ;;fa a = f f a + f a ( ; a): Implication (18) is actually Trahan's Theorem [2]. Corollary 2 (Mn -condition). By applying M -condition on function n we get Mn -condition. Thus, the following implication is true: 0n (a) n (b) < 0 =) (9 2 (a; b)) 0n ( ) = 0: (21) The re ection of the previous relation on the original function f is given by implication: ( )

0

( )+

0

( )

( )

( )

( )

( )

( )

( )

( )

( )

0

( )

( )+

3

2

00

0

( )

2

0

( )

( )+2

6

0

( )

( )

( )

4

3

( )

0

( )

( )+2

0

00

( )

3

( )

6

1

f n (a) Rnf (a; b) < 0 =) (

;

9 2 (a; b)

(22)

+1)

nP ;

+1

(j) f ;f a = (1 ; nj ) f j a ( ; a)j; + f ;a j ( )

( )

1

( )

0

1

!

=2

( )+(

n; f a n 1)

0

( )

:

For n = 1; 2; 3 we get the corresponding implications: f 00 (a) Rf (a; b) < 0 =) (9 2 (a; b)) f ( ) ; f (a) = f 0 ( );

;a f (a) = f 0 ( ) + f 0 (a) f 000 (a) Rf (a; b) < 0 =) (9 2 (a; b)) f () ; ;a 2 2

3

and

f lV (a) Rf (a; b) < 0 =) (9 2 (a; b)) f ;;af a = f f a + f a ( ; a): ( )

4

( )

0

( )+2 3

0

( )

00

(23) (24)

( )

6

(25)

13

Some mean value theorems in terms of an in nitesimal function

Implication (23) provides one more sucient condition that the conclusion of the Flett's Theorem is true.

Taylor's theorem. On the basis of Taylor's expansion of the function f with the remainder in Peano's form: m P

f (a + x) ; f (a) =

k

=1

f (k) a (x)k + o;(x)m ; k ( )

!

using the equality f k a = 0k; (a), k = 1; . . . ; m ^ 0 (a) = f 0 (a), we get the other formulation of the Taylor's expansion: (k)

def

( )

!

0

1

f (a + x) ; f (a) =

m P

k

;

0k; (a) (x)k + o (x)m : 1

=1

Hence, we get a geometric interpretation of the coecient of the k order of the Taylor's expansion as the slope of the tangent function k; at the point A[a; 0]. th

1

REFERENCES [1] T. Flett, A mean value theorem, Math. Gaz. 42 (1958), 38{39. [2] D. Trahan, A new type of mean value theorem, Math. Mag. 39, 5 (1966), 264{268. [3] D. Adamovic, Problem 309, Mat. Vesnik 11(21) (1974), problem section. [4] J. Malesevic, O jednoj teoremi G. Darboux-a i teoremi D. Trahan-a, Mat. Vesnik 4(13)(32) (1980), 304{312. (received 25.04.1997, in revised form 10.06.1999.) Faculty of Electrical Engineering, University of Belgrade, Yugoslavia E-mail : [email protected]

UDK 517.28 research paper

originalni nauqni rad

SOME MEAN VALUE THEOREMS IN TERMS OF AN INFINITESIMAL FUNCTION Branko J. Malesevic Abstract. In this paper we survey some mean value theorems in terms of an in nitesimal

function.

1. Introduction

Let f : [a; b] ! R be a function which is dierentiable in a segment [a; b] and dierentiable arbitrary number of times in a right neighbourhood of the point x = a. According to the following relation: y = f (x) ; f (a) = f 0 (a)(x ; a) + (x)(x ; a); where limx!a (x) = limx!a (x) = 0, let us introduce the function with: f x ;f a 0 x;a ; f (a); x 2 (a; b] (x) = (1) 0; x=a It is true that: f 0 (x) = f (x) ; f (a) ; f 0(a) = f (x) ; f (a) ; f (a)(x ; a) = R (a; x) (x 6= a); +

1

(

def

)

( )

1

x;a

1

2

x;a

x;a

where Rf (a; x) is the remainder of the second order in Taylor's formula. Then, 2

(2)

(x) ; (a) = lim f (x) ; f (a) ; f 0 (a)(x ; a) 0 (a) = xlim x!a !a x;a (x ; a) 0 0 f (x) ; f (a) 1 00 = xlim !a 2(x ; a) = 2 f (a): 1

1

1

2

We generalize this by induction as:

n (a) = n +1 1 f n (a); (

)

(

+1)

1

for an arbitrary n 2 N. The given formula is also a corollary of [3]. AMS Subject Classi cation : 26 A 24

9

(3)

10

B. J. Malesevic

Let us introduce analogously:

(x) =

def

2

1 x ;1 a x;a (

)

; 0 (a); x 2 (a; b]

( )

1

0;

(4)

x = a:

It is true that

(a) 0 (x) = (xx) ; ; a ; (a) f (x) ; f (a) ; f 0(a)(x ; a) ; f 00 (a)(x ; a) Rf (a; x) = (x ; a) (x 6= a); = (x ; a) 1

1

2

1

1

2

3

2

2

2

where Rf (a; x) is the remainder of the third order in Taylor's formula. According to the formula (3) the following formula for n derivative is true: n (a) = n +1 1 n (a) = (n + 1)(1 n + 2) f n (a): Next, let us introduce a sequence of functions: 3

th

(

)

(

2

+1)

(

k (x) =

def

k x ;k a x;a (

)

( )

0; For the function k let us assume: +1

; 0k (a); x 2 (a; b]

(k = 1; 2; . . . ):

x = a;

kn (a) = (n + 1)(n + 12) (n + k) f n k (a) (

+2)

1

)

(

(5)

(n 2 N)

+ )

and k (x) = Rkf (a; x)=(x ; a)k (x 6= a), where Rkf (a; x) is the remainder of (k +1) order in Taylor's formula. Then, according to the formula (3) and inductive assumption for k, we have: kn (a) = n +1 1 kn (a) = (n + 1)(n + 2)1 (n + k + 1) f n k (a) (6) and (x) = k (x) ; k (a) ; 0 (a) +1

+1

th

(

(

)

+1)

(

+ +1)

+1

k

+1

k x;a Rf (a; x) ; 0 ; k f k (a) (x ; a)k = k (x ; a)k

Rf (a; x) = (xk; a)k (x 6= a); (7) is the remainder of (k + 2) order in Taylor's formula. 1

+1

( +1)

+1

( +1)!

+2

+1

where Rkf

+1

th

+2

2. Mean value theorems Flett's theorem. We determine the conditions which are sucient for the existence of the point 2 (a; b) in which the function has an extreme value 1

11

Some mean value theorems in terms of an in nitesimal function

and the equality 0 ( ) = 0 is true. The re ection of the previous equality on the function f is Flett's equality: f 0 ( ) = f ( ) ; f (a) : (8) 1

;a

Namely, Flett [1] has proved that the condition: f 0 (a) = f 0 (b) (9) is sucient for the existence of the point 2 (a; b) so that (8) is true. Besides the previous condition, it is shown that the conclusion of the Flett's Theorem is true under the other conditions, as well. The following statement is true. Theorem. If one of the following two conditions is true: 0 (b) (b) < 0; T -condition; 1

1

(10)

1

and

0 (a) (b) < 0 M -condition; then there exists a point 2 (a; b), such that the following holds: ; f (a) : f 0 ( ) = f () ; a 1

1

(11)

1

(12)

If both of the conditions (10) and (11) are ful lled, then there exists at least two points i 2 (a; b), such that the following holds:

; f (a) f 0 (i ) = f (i ) ; a

(i = 1; 2):

i

(13)

From the condition of the corollary 3 of the theorem 2 [4]: 0 (a)[ (b) ; (a)] < 0; and 0 (b)[ (b) ; (a)] < 0; (14) we obtain the T and M conditions: 0 (a) (b) < 0; and 0 (b) (b) < 0; (15) respectively. According to the second part of the stated corollary for r = a the point 2 (a; b) exists, for which 0 ( ) = 0, i.e. f 0 ( ) = f ;;af a ; while for r = b the point 2 (a; b) exists, for which 0 ( ) = 0, i.e. f 0 ( ) = f ;;af a . If both of the stated conditions (10) and (11) are ful lled (T -M -condition), according to the rst part of the stated corollary, then two points i 2 (a; b) exist, for which 0 (i ) = 0, i.e. f 0 (i ) = f ;;fa a (i = 1; 2). We give extensions of the previous conditions on function n , with the suitable conclusions. Proof.

1

1

1

1

1

1

1

1

1

2

1

2

1

( 2)

2

1

( i)

i

( )

( )

2

1

( 1)

1

1

1

1

1

1

( )

12

B. J. Malesevic

Corollary 1 (Tn -condition). By applying T -condition on function n we get Tn -condition. Thus, the following implication is true: 0n (b) n (b) < 0 =) (9 2 (a; b)) 0n ( ) = 0: (16) The re ection of the previous relation on the original function f is given by implication: 1

nP ;1

n;j f (j) a (b ; a)j ; + f b n; f a ; f b ;f a Rf (a; b) < 0 n n j n b;a j nP ; ; (j) =) 9 2 (a; b) f ;;fa a = (1 ; nj ) f j a ( ; a)j; + f nn; f a j ( )

0

1

( )+(

0

1)

( )

( )

( )

+1

!

=2

( )

1

( )

( )

0

1

( )+(

0

1)

!

=2

(17)

:

( )

For n = 1; 2; 3 we get the corresponding implications: ; ; 0 f (a) ; f bb;;af a f 0 (b) ; f bb;;af a > 0 =) (9 2 (a; b)) f ;;af a = f 0 ( ); (18) ;f b f a f b ; f a f ; f a f f a f ; b;a R (a; b) < 0 =) (9 2 (a; b)) ;a = (19) and ;f a (b ; a) + f b f a ; f bb;;af a Rf (a; b) < 0 (20) =) (9 2 (a; b)) f ;;fa a = f f a + f a ( ; a): Implication (18) is actually Trahan's Theorem [2]. Corollary 2 (Mn -condition). By applying M -condition on function n we get Mn -condition. Thus, the following implication is true: 0n (a) n (b) < 0 =) (9 2 (a; b)) 0n ( ) = 0: (21) The re ection of the previous relation on the original function f is given by implication: ( )

0

( )+

0

( )

( )

( )

( )

( )

( )

( )

( )

( )

0

( )

( )+

3

2

00

0

( )

2

0

( )

( )+2

6

0

( )

( )

( )

4

3

( )

0

( )

( )+2

0

00

( )

3

( )

6

1

f n (a) Rnf (a; b) < 0 =) (

;

9 2 (a; b)

(22)

+1)

nP ;

+1

(j) f ;f a = (1 ; nj ) f j a ( ; a)j; + f ;a j ( )

( )

1

( )

0

1

!

=2

( )+(

n; f a n 1)

0

( )

:

For n = 1; 2; 3 we get the corresponding implications: f 00 (a) Rf (a; b) < 0 =) (9 2 (a; b)) f ( ) ; f (a) = f 0 ( );

;a f (a) = f 0 ( ) + f 0 (a) f 000 (a) Rf (a; b) < 0 =) (9 2 (a; b)) f () ; ;a 2 2

3

and

f lV (a) Rf (a; b) < 0 =) (9 2 (a; b)) f ;;af a = f f a + f a ( ; a): ( )

4

( )

0

( )+2 3

0

( )

00

(23) (24)

( )

6

(25)

13

Some mean value theorems in terms of an in nitesimal function

Implication (23) provides one more sucient condition that the conclusion of the Flett's Theorem is true.

Taylor's theorem. On the basis of Taylor's expansion of the function f with the remainder in Peano's form: m P

f (a + x) ; f (a) =

k

=1

f (k) a (x)k + o;(x)m ; k ( )

!

using the equality f k a = 0k; (a), k = 1; . . . ; m ^ 0 (a) = f 0 (a), we get the other formulation of the Taylor's expansion: (k)

def

( )

!

0

1

f (a + x) ; f (a) =

m P

k

;

0k; (a) (x)k + o (x)m : 1

=1

Hence, we get a geometric interpretation of the coecient of the k order of the Taylor's expansion as the slope of the tangent function k; at the point A[a; 0]. th

1

REFERENCES [1] T. Flett, A mean value theorem, Math. Gaz. 42 (1958), 38{39. [2] D. Trahan, A new type of mean value theorem, Math. Mag. 39, 5 (1966), 264{268. [3] D. Adamovic, Problem 309, Mat. Vesnik 11(21) (1974), problem section. [4] J. Malesevic, O jednoj teoremi G. Darboux-a i teoremi D. Trahan-a, Mat. Vesnik 4(13)(32) (1980), 304{312. (received 25.04.1997, in revised form 10.06.1999.) Faculty of Electrical Engineering, University of Belgrade, Yugoslavia E-mail : [email protected]