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ARTICOLE SI NOTE DE MATEMATICA
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A generalisation of a problem from Gazeta Matematic˘a

1
Leonard Giugiuc and Alexandru Daniel Pˆırvuceanu 2

The following is the problem 27767 from Gazeta Matematic˘a, No. 11/2019:

27767. Let f : [0, 1] → R be a differentiable function with the property that

Z 1
f(x)dx = 0.
0
R 1
0
Prove that there exists c ∈ (0, 1) such that f (c) = −9 3 f(t)dt.
0
Marian Andronache and Dinu S , erb˘anescu, Bucharest

While we were solving this problem, we observed that a more general result holds. What is
interesting is that the official solution published in [2] can be adapted to prove this more general
result. In this article, we aim to present this generalisation, followed by our solution:

Problem. Let a, b ∈ R, a < b. Consider a differentiable function f : [a, b] → R such that

Z b
f(x)dx = 0.
a
Prove that for all x ∈ [a, b] there exists c x ∈ (a, b) such that

x
Z
0
2 f(t)dt = (x − a)(x − b)f (c x ).
a
Solution. If x = a or x = b, then there is nothing to prove, since the conclusion rewrites as
0 = 0.

For x ∈ (a, b), the conclusion is equivalent to proving that there exists c x ∈ (a, b) such that

Z x
2 f(t)dt
0
a = f (c x ).
(x − a)(x − b)

Z y
Consider the function F : [a, b] → R, F(y) = f(t)dt.
a
1
Professor, National ,,Traian” College, Drobeta-Turnu Severin, Romania, leonardgiugiuc@yahoo.com
2
Student, Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest, Romania,
pirvuceanualexandrudaniel@gmail.com.

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