If the function f : I → R is differentiable on the interval I ⊆ R , then for each x,a ∈ I, according to the mean value theorem, there exists a number c(x) belonging to the open interval determined by x and a , and there exists a real number θ (x) ∈]0,1[ such that f (x) − f (a) = (x − a) f (1) (c(x)) and f (x) − f (a) = (x − a) f (1) (a + (x − a)θ (x)). In this paper we shall study the differentiability of the functions c and θ in a neighbourhood of a. Mathematics subject classification (2000): 26A24.

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If the functions f , g : I → R are differentiable on the interval I ⊆ R , then for each x, a ∈ I there exists a real number θ ∈]0, 1[ such that (f (x) − f (a)) g (1) (a + θ(x − a)) = (g (x) − g (a)) f (1) (a + θ(x − a)). In this paper we study the behaviour of the number θ ∈]0, 1[, when x approaches a .

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In this article we study approximation methods for solving bi-criteria optimization problems. Initial problem is approximated by a new one consisting of the second order approximation of feasible set and components of objective function might be initial function, first or second approximation of it. Conditions such that efficient solution of the approximate problem will remain efficient for initial problem and reciprocally are studied. Numerical examples are developed to emphasize the importance of these conditions.