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In a previous article published in the "American Mathematical Monthly," Tucker ("Amer Math Monthly." 1997; 104(3): 231-240) made severe criticism on the Mean Value Theorem and, unfortunately, the majority of calculus textbooks also do not help to improve its reputation. The standard argument for proving it seems to be applying Rolle's theorem to a function like h(x) := f(x) - f(a) - f(b) - f(a)/ b - a(x - a). Although short and effective, such reasoning is not intuitive. Perhaps for this reason, Tucker classified the Mean Value Theorem as "a technical existence theorem used to prove intuitively obvious statements". Moreover, he argued that "there is nothing obvious about the Mean Value Theorem without the continuity of the derivative". Under so unfair discrimination, we felt the need to come to the defense of this beautiful theorem in order to clear up these misunderstandings.

We present some closed form expressions for the Cauchy–Sylvester compound determinant and the determinant of the Adjugate matrix when each minor in their expressions is replaced by some linear functions (including the sum) and products of its elements. [ABSTRACT FROM AUTHOR]

In this note we present a sufficient condition for an upper bound on the rank of a set { A 1 , A 2 , ... , A m } of real n × n symmetric matrices to hold. The condition is based on the rank of a matrix defined by vectors in the image of each matrix, namely A 1 x , A 2 x , ... , A m x , x ∈ R n together with an auxiliary matrix V ∈ R k × m . The result is inspired by some problems on second-order necessary optimality conditions for constrained optimization with quadratic constraints, and we explain and unify previously known results. [ABSTRACT FROM AUTHOR]

Romberg integrals are built in order to accelerate the convergence of sequences of trapezoidal rules for approximating the definite integral of a continuous function f. While every sequence of trapezoidal rules with decreasing step length converges whenever f is continuous, this does not always hold for Romberg integrals. In this note we present a concrete example for which the sequence formed by the diagonal elements of the Romberg table diverges when the number of points used to compute the trapezoidal rules grows too slowly. [ABSTRACT FROM AUTHOR]

We show that Muir’s law of extensible minors, Cayley’s law of complementaries and Jacobi’s identity for minors of the adjugate [Determinantal identitiesLinear Algebra and its Applications52/53 (1983) pp. 769–791] are equivalent. We also show our generalization of Mühlbach/Muir’s extension principle [A generalization of Mühlbach’s extension principle for determinantal identities.Linear and Multilinear Algebra61 (10) (2013) pp. 1363–1376] is equivalent to its previous form derived by Mühlbach. As a corollary, we show that Mühlbach–Gasca–(Lopez-Carmona)–Ramirez identity [A generalization of Sylvester’s identity on determinants and some applications.Linear Algebra and its Applications66 (1985) pp. 221–234/On extending determinantal identities.Linear Algebra and its Applications132 (1990) pp. 145–162] is equivalent to its generalization found by Beckermann and Mühlbach [A general determinantal identity of Sylvester type and some applications.Linear Algebra and its Applications197,198 (1994) pp. 93–112]. [ABSTRACT FROM PUBLISHER]