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Rev. Acad. Canar. Cienc., XIII (Núms. 1-2-3), 35-38 (2001) (publicado en Julio de 2002)
A PROOF OF THE INTERMEDIATE VALUE THEOREM ADAPTED
TO POLYNOMIALS
J. M. ALMIRA AND M. JIMENEZ
l. INTRODUCTION
Rolle (1652-1719) published his famous theorem in 1691 (see [5, p. 504]).
Theorem l. (Rolle} Let us assume that f E C[a,b] has derivative on (a,b). If f(a) =
f(b) =O then !'(8) =O for sorne 8 E (a, b).
After this, many mathematicians used the following variation of the theorem:
Theorem 2. {Mean Value Theorem) Let us assume that f E C[a, b] has derivative
on (a; b). Then there exists a certain value 8 E (a, b) such that
f (b) - f (a) = !'(8)
b-a
Lagrange (1736-1813) tried to prove the mean value theorem, but he failed, since his
point of view was completely wrong in the sense that he tried to make Calculus in a
purely algebraic way (see [1], [3], [4]). The first mathematician which sistematically used
the above result was Cauchy (1789-1857), who also 'proved' the following generalization:
Theorem 3. (Cauchy's Mean Value Theorem) Let us assume that f, g E C[a, b]
both have derivative on (a, b), and g(a) =f:. g(b). Then there exists a certain value 8 E (a, b)
such that
f(a) - f (b) !'(8)
=
g(a) - g(b) g'(8)
A proof of Rolle's theorem is as follows: Firstly we use that f attains an extreme value
at sorne e E (a, b) (whenever it is not constant). Hence f'(c) = O, since it is a derivable
function. On the other hand, if f is constant then f' is identically zero.
Of course, it is easy to note that we have used the existence of extreme points for
continuous functions defined on intervals. This is a theorem which was formally proved
by Weierstrass (much more after than Rolle's theorem) and in a modern perspective,
the key for the proof is that intervals are compact sets. The same can be applied to
the famous Bolzano's theorem (that a continuous function which changes the sign must
vanish at sorne point). The conclusion is that the topology of real numbers was not an
easy matter and, in fact, was the key for the simplification of many results from the
foundations of Analysis.
Cauchy's main objective was to introduce the rigor in the proofs of general properties
of continuous functions. He said it was neccesary to give exact proofs for continuous
Research partially supported by Junta de Andalucía, Grupo de Investigación "Aproximación y Métodos
Numéricos" FQM-0178 and PB-94; email: jmalmira@ujaen.es, mjimenez@ujaen.es.
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functions of many results that people know to be true for algebraic functions (see [5, p.
1252]).
We have sorne doubts with respect to the question if mathematicians of that period
had the proofs of sorne elementary facts of Mathematical Analysis, for the class of real
algebraic polynomials. For example: what about the intermediate value theorem? In this
note we give proofs adapted for the space of polynomials of sorne of these results. We
believe our proofs could be understood by Cauchy's contemporaries.
2. THE PROOFS
Lemma 4. {Increasing Function Theorem, for Polynomials {IFTP)) Let us
assume that p(x) E JR[x] satisfies p'(x) ~O for all x E (a, b) (p'(x) ~O for all x E (a, b),
respectively). Then p is an increasing function (a decreasing function, respectively) on
[a,b].
Proof. We use the geometric concept of integral to prove that J: xndx = ~:; - ~n_:; (a
fact that was already proved by Fermat in 1636, see [5, p. 466]). Hence the relation
p(x) - p(a) =fax p'(t)dt holds trivially true for the class of polynomials. If p'(t) is greater
(or equal) than zero on [a, b] then x1 , x2 E [a, b], x1 ~ x2 implies
p(x2) - p(xi) = ¡x2 p'(t)dt ~ O lx1
Hence p is an increasing function in such a case. o
Lemma 5. {Intermedia~· Value Theorem for monotone functions {IVT)) Let
us assume that f E C[a, b] is monotone. Then there ex1st m, ME lR, m < M, such that
f([a,b]) = [m,M] .
Proof. We assume without loss of generality that f is increasing and e E (!(a), f(b)) .
Now, we set A1 = {x E [a,b]: f(x) <e} and A2 = {x E [a,b]: f(x) >e}. They are both
non empty bounded sets, so that we can define the quantities a = sup A1 and /3 = inf A2 .
It follows from the monotonicity off that a~ /3. Take {xn}::'=o e Ai, {Yn}::'=o e A2 such
that limxn =a and limyn = /3. Then f(a) = limf(xn) ~e~ limf(Yn) = f(/3), so that
a = /3 clearly implies that f(a) = c. On the other hand, if a i= /3 then it is clear that
(a+ /3)/2 E [a, b] \ (A1 U A2), so that f((a + /3)/2) =c. O
We must observe that the notation of supremun, etc, were not known at that time. In
fact, people from Cauchy's time were notable to use the axiom of supremun (that we use
now easily). In his famous paper, Bolzano used (without proof) the following claim, which
is equivalent to the axiom of supremun: lf a property P is satisfied by all numbers less or
equal to a certain number x, and the property does not hold true for all (real} numbers,
then there exists a number, which is the greatest in the set of (real) numbers M such that
P holds true for all y < M. He was never able to prove this property. From our point of
view, the proof we have presented here, could be though -but not stated with complete
rigor- at that time.
Theorem 6. {Intermediate Value Theorem, for Polynomials (IVTP)) Let
p(x) E JR[x] be a polynomial with coefficients in JR, and let a, b E lR, a < b be fixed. Then
there exist m, ME JR, m < M, such that p([a, b]) = [m, M].
Proof. lt follows from the algorithm of division of polynomials (Euclid) that a polynomial
p(x) = a0 + a1x + · · · anxn has at most n zeros. This implies that its derivative (which
is another polynomial) has also a finite number of zeros (at most n - 1), so that (using
Lemma 4), we can decompose the interval [a, b] as a finite union of closed intervals [a, b] =
11 U 12 U ... Uh, where l. = [ts-1> t.], a = t 0 < t1 < ... < tk = b and P11. is a monotone
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function for each s :S k. It follows from Lemma 5 that p(I,) = J, = [m., M,j is a closed
interval for all s. Furthermore, t, E Js+1 n J,, so that J,+l n J, i= 0 for all s. This means
that J,+l U J, is another closed interval and, repeating the argument severa! times, we
obtain that p([a, b]) = J1 U J2 U ... U Jk = [m, M] is a closed interval. D
Of course, there are elementary proofs of the increasing function theorem for derivable
functions (see [6]),
Theorem 7. {lncreasing .Function Theorem {IFT)) Let us assume that f E C[a, b]
is differentiable on (a, b) and f'(x) ~ O for all x E (a, b) (f'(x) :S O for all x E (a, b),
respectively). Then f is an increasing function (a decreasing function, respectively) on
[a, b],
but they use Cantor's principie of nested intervals (it asserts that the intersection of a
nested sequence of closed intervals whose diameters approach to zero, is a point), which
is the key idea for the bisection method (hence for a proof of Bolzano's theorem and, as
a consequence, a proof of the intermetiate val u e theorem for continuous functions).
Another proof of the IVTP which could be thought in Cauchy's times is as follows:
We can use the Fundamemal Theorem of Algebra (firstly proved by Gauss in 1799, see
[1]) to decompose the polynomial p(x) E JR[x] as a product of\irreducible factors:
n m
i=l j=l
where the polynomials qj(x) are quadratic factors without real zeros, so that the sign of
ITT=i qj(x)m; is constant on the real line. Hence the changes of sign in the interval [a, b]
are caused by the linear factors a;x + b;. This clearly implies that Bolzano's Theorem
holds for the class of polynomials. The intermediate value theorem is now a corollary of
Bolzano's theorem. This proof also serves for polynomials in the field of algebraic real
numbers, or any other really closed field. Note that to use this focus, we must firstly
prove the fundamental theorem of algebra, without the use of IVTP.
Rolle's theorem and the Mean Value Theorem have now a proof for the class of polynomials
as corollaries of the IVTP. Other consequences are the following theorems:
Theorem 8. If p(x) = a0 + ... + an_1xn-I + xn is a polynomial of odd degree, then it has
a real root.
Proof. It suffices to note that
and use the IVTP. D
Theorem 9. Let a< b be two real numbers and let p(x) be a polynomial with real coefficients.
Then p is uniformly continuous on [a, b].
Proof. Let [m, M] = p'([a, b]) and set C = max{jmj, jMj}. Then
jp(x) - p(y)j = jp'(O)llx - YI :S Cjx - YI
holds for all x, y E [a, b]. D
Of course, we must recall that many results from a first year Calculus course have trivial
proofs for the set of polynomials. For example, the chain rule can be checked by direct
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computation, since the formula (q(zr)' = (q(zr- 1q(z))'
q(z)n-1q'(z) = nq(z)n-1q'(z ), n = 1, 2, ... is clear, and
(n - l) q( z )n-•q1(z )q(z) +
(p o q)'(z) = (~ akq(z)k)' = ~ kakq(z)k-1q'(z) = p'(q(z)) q'(z).
We are convinced that sorne of these elementary proofs where already known at the time
of Cauchy. Perhaps he was thinking how to extend these results to the class of continuous
functions, when he decided to write his Cours d'Analyse [2]. If he was completely right or
not, is not the main question. From our point of view, the most important thing is that
Cauchy's work had a deep influence in other mathematicians for a long period of time,
and it was a great contribution for the introduction of rigor in Mathematical Analysis.
The main goal of this note was nothing but to fill the gap caused by Cauchy's claim about
the existence of special proofs of sorne elementary facts from Calculus, adapted to the
space of real algebraic polytfomials.
REFERENCES
[1] Boyer, C. A History o/ Mathemtstics. John Wiley & Sons, (1968)
[2] Cauchy, A. L. Cours d'analyse algébrique (1821), in Ouvres Completes, Gauthier-Villars, Paris.
[3] Duran, A. J. Historia, con personajes, de los conceptos del cálculo, Alianza Editorial. Madrid.
(1996)
[4] Grattan-Guinnes, l. From the Calculus to Set Theory 1690-19101lurckworth (1980) (translated
into Spanish by Alianza Ed. in (1996)).
[5] Kline, M., Mathematical Thought from Ancient to Modem Times, Oxford University Press (1972).
[6] Tucker, T. W., Rethinking rigor in Calculus: the role of the Mean Value Theorem, Amer. Math.
Monthly 104 (1997) 231-240.
J. M. Almira, M. Jiménez.
Departamento de Matemáticas. Universidad de Jaén.
E. U. P. Linares, C/ Alfonso X el Sabio, 28.
23700 Linares (Jaén) SPAIN.
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