General Mathematics Vol. 12, No. 3 (2004), 53–59
Continuity properties relative to the
intermediate point in a mean value theorem
Emil C. Popa
Abstract
This article studies the existence and the value of the limit lim
x↑a
x−a
(cx − a)α
where cx is the intermediate point in a mean value theorem.
2000 Mathematical Subject Classification: 26A24
Keywords: mean value theorem, intermediate point.
1◦ . We shall consider the following mean value theorem:
Theorem 1. Let us suppose that f : [a, b] → R is continuous on [a, b] and
derivable on (a, b). If A(a, f (a)), B(b, f (b)), then for each M (x, y) ∈ AB,
x 6∈ [a, b], exists a point c ∈ (a, b), such that:
y − f (c)
= f 0 (c).
x−c
53
54
Emil C. Popa
The geometric significance of this theorem is the fact that from any
point M (x, y) ∈ AB where x 6∈ [a, b] one can draw a tangent to the graphic
representation of f (see [3]).
In [1], [3], [4], there are studies for the classical mean value theorems,
under some hypothesis, the existence and the value of the limit lim c − a ,
b→a b − a
c = c(b) being the intermediate point in the respective mean value theorem.
In the following we shall prove that:
lim
x↑a
(1)
lim
x↑a
x−a
= 0, for α < 2, and
(cx − a)α
x−a
f 00 (a)
f (b) − f (a)
=
, mAB =
,
0
2
2[f (a) − mAB ]
b−a
(c − a)
where “c” is the intermediate point from Theorem 1 with the additional
conditions that f should be twice derivable, f 00 < 0 on [a, b].
We can now prove the following:
Theorem 2. If f : [a, b] → R is two times differentiable on [a, b] with
f 00 < 0 on [a, b] and A(a, f (a)), B(b, f (b)) then for each point M (x, y),
M ∈ AB, x ∈ (−∞, a), there is an unique point c ∈ (a, b) such that:
y − f (c)
= f 0 (c).
x−c
Proof. We consider that exists a < c1 < c2 < b such that:
y − f (c1 ) = (x − c1 )f 0 (c1 )
y − f (c2 ) = (x − c2 )f 0 (c2 )
We have:
f 0 (c2 ) <
f (c1 ) − f (c2 )
= f 0 (ξ) < f 0 (c1 )
c1 − c2
Continuity properties relative to the intermediate point...
55
with ξ ∈ (c1 , c2 ).
Therefore
(c1 − c2 )f 0 (c2 ) > (x − c2 )f 0 (c2 ) − (x − c1 )f 0 (c1 ),
(c1 − x)f 0 (c2 ) > (c1 − x)f 0 (c1 ),
f 0 (c2 ) > f 0 (c1 )
in contradiction with f 0 (c1 ) > f 0 (c2 ).
Hence exists an unique point c ∈ (a, b) with
y − f (c)
= f 0 (c).
x−c
2◦ . Further we prove that the following theorem is valid.
Theorem 3. Let f : [a, b] → R such that f 00 exists and f 00 < 0 on [a, b]. If
e
c ∈ (a, b) is the unique point having the property
f 0 (e
c) =
f (b) − f (a)
,
b−a
then:
i) whatever is c ∈ (a, e
c), the tangent to the graphic representation of f
in the point (c, f (c)) cuts AB in M (xc , y) with xc < a.
ii) whatever is c ∈ (e
c, b), the tangent to the graphic representation of f
in the point (c, f (c)) cuts AB in N (xc , y) with xc > b.
Proof.
i) The tangent to the graphic representation of f in the point
(c, f (c)) cuts AB in a point having the abscissa:
xc =
cf 0 (c) − f (c) + f (a) − a · mAB
f 0 (c) − mAB
56
Emil C. Popa
where we have denoted
mAB =
If
Ψ(t) =
f (b) − f (a)
.
b−a
tf 0 (t) − f (t) + f (a) − a · mAB
, t ∈ [a, e
c),
f 0 (t) − mAB
we have:
Ψ0 (t) =
f (t) − [f (a) + (t − a)mAB ] 00
f (t).
[f 0 (t) − mAB ]2
But Ψ0 (t) < 0 on (a, e
c), Ψ(a) = a and from Ψ(a) > Ψ(c) it results a > xc .
In a similar way we prove the second part of the above theorem.
Remark 1. Because f 0 is strictly decreasing on [a, b] it follows the unicity
of e
c and likewise that f 0 (a) − mAB 6= 0.
c ∈ (a, b) is the
Theorem 4. If f : [a, b] → R with f 00 < 0 on [a, b] and e
unique point having the property
f 0 (e
c) =
f (b) − f (a)
b−a
then whatever is x ∈ (−∞, a), there exists exactly one point c ∈ (a, e
c) such
that the tangent to the graphic representation of f in the point (c, f (c)) cuts
AB in M (x, y).
Proof. The existence and the uniqueness of c ∈ (a, b) is given by Theorem
2. If c ∈ (e
c, b) then according to Theorem 3 ii), the tangent to the graphic
representation of f in (c, f (c)) cuts AB in N (xc , y), xc > b. Hence c ∈ (a, e
c).
3◦. Further let f : [a, b] → R such that f 00 exists and f 00 < 0 on [a, b]
f (b) − f (a)
and e
c ∈ (a, b) the unique point which f (e
c) =
. We define the
b−a
Continuity properties relative to the intermediate point...
57
function θ(c) = xc , θ : [a, e
c) → (−∞, a] where c and xc have the significance
from Theorem 3. It is clear that θ(a) = a and according to the Theorem 3
and 4 the function θ is bijective.
If c0 ∈ (a, e
c) then
xc =
cf 0 (c) − f (c) + f (a) − a · mAB
f 0 (c) − mAB
and when c → c0 we have
xc →
c0 f 0 (c0 ) − f (c0 ) + f (a) − a · mAB
= xc0 .
f 0 (c0 ) − mAB
So θ(c) → θ(c0 ) and θ is continuous on (a, e
c).
In addition, when c → a it is evidently that
af 0 (a) − a · mAB
xc =
= a.
f 0 (a) − mAB
Hence θ(c) → θ(a) and θ is also continuous in a, and on [a, e
c).
Because θ is bijective and continuous it follows that θ−1 is also continuous
on (−∞, a]. We denote θ−1 (x) = cx and when x → a, θ−1 (x) → θ−1 (a) that
is cx → a.
Finally we shall prove the following:
Theorem 5. Under the hypothesis of the Theorem 2 we have:
lim
x↑a
x−a
= 0,
cx − a
and
lim
x↑a
x−a
f 00 (a)
=
.
2[f 0 (a) − mAB ]
(cx − a)2
58
Emil C. Popa
Proof. From the relation
y − f (cx )
= f 0 (cx )
x − cx
where y = f (a) + (x − a)mAB , we obtain:
f (a) − f (cx ) + (x − a)mAB = (x − a)f 0 (cx ) + (a − cx )f 0 (cx )
which implies:
x−a
=
cx − a
(2)
f (cx ) − f (a)
cx − a
f 0 (cx ) − mAB
f 0 (cx ) −
and
x−a
(cx − a)f 0 (cx ) − f (cx ) + f (a)
1
=
· 0
.
2
2
f (cx ) − mAB
(cx − a)
(cx − a)
For x → a we have cx → a and
f 0 (cx ) → f 0 (a),
f (cx ) − f (a)
→ f 0 (a).
cx − a
According to (2) we find
lim
x↑a
x−a
= 0.
cx − a
If we use l, Hôspital, s rule, we have:
(cx − a)f 0 (cx ) − f (cx ) + f (a)
1
= f 00 (a)
2
cx →a
2
(cx − a)
lim
and so:
x−a
f 00 (a)
lim
=
.
x↑a (cx − a)2
2[f 0 (a) − mAB ]
Remark 2. It is clear that:





0,
α<2
00
x−a
f (a)
lim
, α=2 .
0
α =
2[f
(a)
− mAB ]
x↑a (cx − a)




−∞,
α>2
Continuity properties relative to the intermediate point...
59
References
[1] Jakobson, B., On the mean value theorem for integrals, The American
Mathematical Monthly, vol. 87, 1982, p. 300-301.
[2] Muntean, I., Extensions of some mean value theorems, “Babeş-Bolyai”
University, Faculty of Mathematics, Research Seminar, Seminar on
Mathematical Analysis, Preprint, no. 7, 1991, p. 7-24.
[3] Popa, E. C., On a mean value theorem (in romanian), MGB, nr. 4,
1989, p. 113-114.
[4] Popa, E.C., On a property of intermediary point “c” for a mean value
theorems (in romanian), MGB, nr. 8, 1994, p. 348 - 350.
“Lucian Blaga” University of Sibiu
Faculty of Sciences
Department of Mathematics
Str. Dr. I. Raţiu, no. 5-7,
550012 - Sibiu, Romania
E-mail: emil.popa@ulbsibiu.ro