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# A converse of the mean value theorem made easy

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## Abstract

The aim of this article is to discuss some results about the converse mean value theorem stated by Tong and Braza [J. Tong and P. Braza, A converse of the mean value theorem, Amer. Math. Monthly 104(10), (1997), pp. 939–942] and Almeida [R. Almeida, An elementary proof of a converse mean-value theorem, Internat. J. Math. Ed. Sci. Tech. 39(8) (2008), pp. 1110–1111]. We provide a simpler proof of the results of Tong and Braza and finally a generalization is given.
A CONVERSE OF THE MEAN VALUE THEOREM MADE EASY
CRISTINE L MORTICI
Abstract. The aim of this paper is to discuss some results about the converse
mean value theorem stated by Tong and Braza [A converse of the mean value
theorem Amer. Math. Monthly 104(10) (1997) 939-942] and Almeida [An
elementa ry proof of a conv erse mean-value theorem Internat. J. Math. Ed .
Sci. Tech. 39: 8, 1110-1111]. We provide a simpler proof of the results of Tong
and Braza and nally a gener alization is given.
Tong and Braza [2] proved the following converse of the mean value theorem.
Theorem 1 (Tong and Braza). Let f(x) be a function continuous on [a; b] and
di¤erentiable on (a; b) and let c be a given point in (a; b). Then
(1) Weak Form: If f
0
(c) is not a total extremum value on (a; b), i.e., f
0
(c) 6=
supff
0
(x) j x 2 (a; b)g and f
0
(c) 6= infff
0
(x) j x 2 (a; b)g, then there is some
subinterval (a
1
; b
1
) (a; b) such that f
0
(c) = (f(b
1
) f(a
1
))=(b
1
a
1
).
(2) Strong Form: If f
0
(c) is not a local extremum value of f
0
(x) on (a; b) and
if c is not an accumulation point of the set A
c
= fx 2 (a; b) j f
0
(x) = f
0
(c)g,
then there is a subinterval (a
1
; b
1
) (a; b) such that a
1
< c < b
1
and f
0
(c) =
(f(b
1
) f(a
1
))=(b
1
a
1
).
The proof given in [2] is quite complicated and this fact determined Almeida [1]
to wonder whether it contains a gap. Observing that Tong and Braza [2] used a
condition of the form
(0.1) lim
x;y !c
f (y) f (x)
y x
= f
0
(c) ;
Almeida [1] considered the di¤erentiable function
f (x) =
x
2
sin
1
x
; x 6= 0
0 ; x = 0
with f
0
(0) = 0 and proved that
lim
k!1; k odd
f (y
k
) f (x
k
)
y
k
x
k
= 2 sin
1
2
6= f
0
(0) ;
where x
k
= 1= (k) ; y
k
= 1= (k) + 1=2 (k)
2
; for k = 1; 3; 5; ::: . Remark that in
this example we have 0 < x
k
< y
k
:
This is indeed a counterexample for (0.1), but the proof of Tonga and Braza can
be modi…ed such that x; y considered in (0.1) satisfy x < c < y: Now, although
(0.1) is not true in general, as Almeida [1] proved, its correct version is the following
1991 Mat hematics Subject Classication. 26A06; 26 A24.
Key words and phrases. mean value theorem; intermediate point ; di¤erentiability.
1
2 CRISTINEL MORTIC I
Theorem 2. If f : (a; b) ! R is di¤erentiable at c 2 (a; b) ; then
lim
x;y !c; x<c<y
f (y) f (x)
y x
= f
0
(c) :
Proof. As we have the identity
f (y) f (x)
y x
=
y c
y x
f (y) f (c)
y c
+
c x
y x
f (x) f (c)
x c
;
we deduce that
f (y) f (x)
y x
=
f (y) f (c)
y c
+ (1 )
f (x) f (c)
x c
;
where = (y c) = (y x) 2 (0; 1) : Thus
min
f (x) f (c)
x c
;
f (y) f (c)
y c
f (y) f (x)
y x
max
f (x) f (c)
x c
;
f (y) f (c)
y c
and the theorem is proved.
Next we give a much simpler proof of Theorem 1.
Proof of Theorem 1. We de…ne the function h : (a; b) ! R by the formula
h (x) = f (x) f
0
(c) x
and we claim that it is not injective. If assume by contrary that h is injective, then h
is strictly monotone, say strictly increasing, since h is derivable. As a consequence,
for every x 2 (a; b) ; we have h
0
(x) 0; or equivalently, f
0
(x) f
0
(c) for every
x 2 (a; b) : Thus f
0
(c) is a total minimum value on (a; b), a contradiction.
As h is not inj ective, one can nd a
1
6= b
1
in (a; b) such that h (a
1
) = h (b
1
) ;
which can be rearranjed as
f
0
(c) =
f (b
1
) f (a
1
)
b
1
a
1
:
For the strong form of the theorem, we consider (a
0
; b
0
) (a; b) such that f
0
(c) 6=
f
0
(x) ; for every x 2 (a
0
; b
0
)r fcg (this is possible since c =2 A
c
). Now f
0
(c) is not a
total extremum value of f
0
on (a
0
; b
0
) and the conclusion follows from the previous
weak form.
Finally we prop ose the following generalization.
Theorem 3. Let f(x); g (x) be functions continuous on [a; b] and di¤erentiable on
(a; b); with g
0
> 0 and let c be a given point in (a; b). Then
(1) Weak Form: If f
0
(c)=g
0
(c) is not a total extremum value on (a; b), i.e., f
0
(c)=g
0
(c) 6=
supff
0
(x)=g
0
(x) j x 2 (a; b)g and f
0
(c)=g
0
(c) 6= infff
0
(x)=g
0
(x) j x 2 (a; b)g,
then there is some subinterval (a
1
; b
1
) (a; b) such that f
0
(c)=g
0
(c) = (f (b
1
)
f(a
1
))=(g (b
1
) g (a
1
)).
(2) Strong Form: If f
0
(c)=g
0
(c) is not a local extremum value of f
0
(x)=g
0
(x) on
(a; b) and if c is not an accumulation point of the set A
c
= fx 2 (a; b) j f
0
(x)=g
0
(x) =
f
0
(c)=g
0
(c)g, then there is a subinterval (a
1
; b
1
) (a; b) such that a
1
< c < b
1
and
f
0
(c)=g
0
(c) = (f(b
1
) f(a
1
))=(g (b
1
) g (a
1
)).
A C ONVERSE OF THE M EAN VALUE THEO REM MADE EASY 3
Proof. We de…ne the function j : (a; b) ! R by the formula
j (x) = f (x) g
0
(c) f
0
(c) g (x)
and we claim that it is not injective. If assu me by contrary that j is injective, then
j is strictly monotone, since j is derivable. As a consequence, for every x 2 (a; b) ;
we have j
0
(x) 0; equivalently, f
0
(x) g
0
(c) f
0
(c) g
0
(x) for every x 2 (a; b) : Thus
f
0
(c) =g
0
(c) is a total minimum value on (a; b), a contradiction.
As j is not injective, one can nd a
1
6= b
1
in (a; b) such that j (a
1
) = j (b
1
) ;
which can be rearranjed as
f
0
(c)
g
0
(c)
=
f (b
1
) f (a
1
)
g (b
1
) g (a
1
)
:
We are convinced that these ideas are suitable to establish other similar results.
References
[1] R. Almeida, An e lementary proof of a converse mean-value theorem, Intern at. J. Math. Ed.
Sci. Tech., 39: 8, 1110-1111.
[2] J. Tong , P. Braza, A converse of the mean value theorem, Amer. Math. Monthly, 104 (1997),
no. 10, 939-942 .
Valahia University of Târgovi¸Ste, Department o f Mathematics, Bd. Unirii 18, 130082
Târgovi¸Ste, Romania