Una simple extensión del teorema de Rolle y su relación con múltiples tasas internas de retorno (TIR)
| Autor | Fernando Gómez-Villarraga |
| Cargo | Public accountant, Universidad de La Salle (Colombia) |
1. Introduction
Rolle’s theorem is attributed to Michel Rolle, and named after this French mathematician. He was a member of the French academy who studied the theory of equations. He stated and proved a preliminary version of the theorem by studying methods to locate and limit the roots of polynomials. Later, Lagrange and Cauchy obtained their mean value theorems applying Rolle’s theorem to appropriate functions. Mean value theorems relate the values of the function with its derivative or integral and are important in the proof of several calculus theorems. Rolle’s theorem is now found in the development of many introductory topics of differential calculus (Martínez de la Rosa, 2008; Pérez-Chavela & Petakos, 2009; Tan, 2011).
The internal rate of return (IRR) is the discount rate that makes the net present value (NPV) of all cash flows from a specific project equal to zero. IRR is the root (or roots in a multiple case) of the NPV expression. IRR is used in the financial decision process to accept or refuse a project. If a specific project has an IRR greater than the minimum required rate of return, then the project should be accepted (the project adds value). While if the IRR is less than the minimum required rate of return, the project should be refused. Multiple IRRs can be obtained in non-conventional (also called non-normal) cash flows. The non-conventional type is a cash flow pattern in which cash flows change the sign more than once (Cissell et al., 1990; Vélez Pareja, 2002).
This paper states and proves an extension of Rolle’s theorem that determines the amount of numbers in which in a given interval, using the characteristics of the function in that interval. This approach has been used to calculate the amount of internal rates of return (IRR) in a non-conventional cash flow.
2. A simple extension of Rolle’s theorem
Let the function be:
(i) continuous on a closed interval
(ii) continuously differentiable on the open interval
(iii) The function is differentiable on the open interval .
(iv) (where is a constant)
(v) (where is a constant)
(vi) and
(vii) There are points in the open interval such that (where is the number of points in the open interval such that ).
(viii) The derivative of at the points is different from zero, ( ).
(ix) The signs of the derivatives of at the points alternate (for ).
Then:
(a) There exist numbers in the open interval such that , as long as:
(a.1) The signs of the lateral right-hand derivative evaluated at , and that of the derivative evaluated at the first point are the same, and
(a.2) The signs of the derivative evaluated at the last point and that of the lateral left-hand derivative evaluated at , are the same.
(b) There exist numbers in the open interval such that , as long as:
(b.1.1) The signs of the lateral right-hand derivative evaluated at , and that of the derivative evaluated at the first point are the same, and
(b.2.1) The signs of the derivative evaluated at the last point and that of the lateral left-hand derivative evaluated at , are different, or
(b.1.2) The signs of the lateral right-hand derivative evaluated at , and that of the derivative evaluated at the first point are different, and
(b.2.2) The signs of the derivative evaluated at the last point and that of the lateral left-hand derivative evaluated at , are the same.
(c) There exist numbers in the open interval such that , as long as:
(c.1) The signs of the lateral right-hand derivative evaluated at , and that of the derivative evaluated at the first point are different, and
(c.2) The signs of the derivative evaluated at the last point and that of the lateral left-hand derivative evaluated at , are different.
Corollary
Let the function be:
(i) continuous on a closed interval
(ii) continuously differentiable on the open interval
(iii) The function is differentiable on the open interval .
(iv) (where is a constant)
(v) (where is a constant)
(vi) there is no point in the open interval such that , (where is the number of points in the open interval such that )
Then, there exists one number in the open interval such that
3. Proof of the simple extension of Rolle’s theorem
If for all in , then for all in ; thus, any number between and can be considered as . To obtain a finite number of points , (condition v).
Since there are points in the interval such that (conditions iii and vii), subintervals can be established given by in the interval , where: are denoted as interior intervals ( subintervals) and as exterior intervals ( subintervals). First, the interior intervals are considered as follows.
As is continuous on the interval (condition i), is also continuous on the interior subintervals . Then, by the extreme value theorem, has an absolute maximum and minimum value in each interior subinterval . The derivative is continuous on the open interval since is continuously differentiable on this interval (condition ii). Thus, is continuous on each interior subinterval . Under these considerations, the intermediate value theorem can be applied to each interior subinterval. The corollary called Bolzano’s theorem is used in this point. Since it is assumed that the signs of the derivatives of at the points alternate (conditions viii and ix), the continuous derivative has values of opposite sign evaluated at the extremes of the interval and , then the derivative has a zero (root) in the interval In addition, since there are no points in each interior subinterval , the graph of is concave upward or concave downward in each interval. Then, has only one extreme value at in each subinterval. Thus, there exist numbers in the...
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