Given a differentiable function f with argument x, its critical points are those values of x, if any, in its domain for which either f′ (x) = 0 or f′ (x) is undefined. The first derivative test is a number line test that tells us, definitively, whether a given critical point, x = c, of f(x) is a local maximum, a local minimum, or neither. The second derivative test is not a number line test, but can also be applied to classify the critical points of f(x). Unfortunately, the second derivative test is, under certain conditions, inconclusive.

There is a rich and seemingly endless source of definite integrals that can be equated to or expressed in terms of Catalan's constant. Denoted by G and defined by

$${\rm{G}} = \sum\limits_{n = 0}^\infty {{{{{\left( { - 1} \right)}^n}} \over {{{\left( {2n + 1} \right)}^2}}} = 1 - {1 \over {{3^2}}} + {1 \over {{5^2}}} \ldots = 0.915\,965\,594 \ldots \,\,,} $$

Geometry is a very interesting, applicable and beautiful part of mathematics. However, geometry is often difficult for students to understand and demanding for teachers to teach [1]. Constructing proofs in geometric problems turns out to be particularly difficult, even for high attaining students [2]. Sometimes, students do not even know where to start when trying to solve these [3].

We tackle an unusual problem that, as far as I know, is not in the standard literature. To state it concisely I use what I call ‘valid sets’. We know that in any three or more positive quantities, only the largest can be half or more of the total value; and is then obviously greater than the sum of all the others. But if the largest is less than half the total of the set, it must be less than the sum of the others; and this is true for every element in this set. I call such a set ‘valid’. For example, the sides of a triangle are valid, as are the face areas of a tetrahedron. Our problem relates to the converse for a tetrahedron: given any four valid quantities, is there always a tetrahedron with those face areas? In this article I answer this by showing that, for any valid set, we can always construct the net of a corresponding tetrahedron. In fact, for any given valid set there is always an infinity of non-congruent tetrahedra with the given face areas. Although in general there are no formulae that give the exact edge lengths of these solids, I show that there are always several distinct infinite subsets that are constructible and whose edge lengths can be determined exactly.

Given a parallelogram, construct squares outwardly on its sides; hereafter we will call this the ‘Thébault configuration’. Our name derives from Thébault’s celebrated first problem, which states that the centres of these newly constructed squares also form a square.

Nowadays there is considerable agreement among educators that learning mathematics fundamentally involves making mathematics [1]. Students learn mathematics while working on tasks that they consider meaningful and worthwhile, and their interest is aroused when they can see the point of what they are being asked to do. Given that learning mathematics involves a process of meaning-making - the use of mathematical language, symbols and representations as learners negotiate ideas – activities should provide students with a variety of challenging experiences through which they can actively construct mathematical meanings for themselves.

In [1] the second author observed that it is possible to have a binary operation * on a set X with the property that two different arrangements of brackets in a given combination x1 * … * xn of elements of X yield the same outcome for all choices of the xj. For example, for the operation of subtraction on the set of real numbers, we have

$$\left[ {a - \left( {b - c} \right)} \right] - d = a - \left[ {b - \left( {c - d} \right)} \right]$$

This article is motivated by, and is meant as a supplement to, the recent paper [1]. That paper proves three geometric characterisations of triangles whose sides are in arithmetic progression, or equivalently triangles in which one of the sides is the arithmetic mean of the other two. More precisely, it gives three geometric contexts in which such triangles appear. In this Article, we supply references for the results in [1] and we provide more proofs of these results. We also add more contexts in which such triangles appear, and we raise related issues for future work. We hope that this will be a source of problems for training for, and for including in, mathematical competitions.

After several years of teaching an introduction to probability and statistics for engineering degree students, my attention has been captured by some variations on the familiar general addition law of probability. Network analysis of components connected in parallel is one of many applications.

Certain ideas recur in many areas of mathematics. One example is groups of symmetries, which appear in the Galois theory of equations and in Lie groups. Lie groups are of great value in physics, where Noether’s theorem enables us to derive a conservation law for every case in which a function known as the Lagrangian is invariant under a one-parameter Lie group. The importance of this approach can be seen from the fact that the laws of the conservation of energy, linear momentum and angular momentum are all outcomes of Noether’s theorem, though they can of course be derived by simpler methods. The full power of Noether’s approach is shown in its applications to quantum field theory, where it can be used to find conserved currents and charges.

A famous theorem of Weierstrass, dating from 1885, states that any continuous function can be uniformly approximated by polynomials on a bounded, closed real interval.

We elaborate the idea behind Markov chain Monte Carlo (MCMC) methods in a mathematically coherent, yet simple and understandable way. To this end, we prove a pivotal convergence theorem for finite Markov chains and a minimal version of the Perron-Frobenius theorem. Subsequently, we briefly discuss two fundamental MCMC methods, the Gibbs and Metropolis-Hastings sampler. Only very basic knowledge about matrices, convergence of real sequences and probability theory is required.

To live to a ripe old age, untroubled by health problems, physical or mental, is an almost universal aspiration. But most people are not so lucky and will likely be in care homes for their final years, with varying levels of disease, disability and dementia. Kinley et al [1] maintain that over a fifth of the population of developed countries die in care homes. Moreover, the financial cost of this end of life care, which is the focus of this paper, can be daunting and require much planning [2]. It was reported in 2017 that, in the UK, care home costs are rising up to twice as fast as inflation [3]. Consequently the question arises about the long term affordability of such care to those having to fund it, a question that ever more people both nationally and globally are having to confront.

The solution set of an equation f(x,y) = g(x,y) over ℝ2 is the set of all pairs of real numbers (x,y) such that the equation holds.

This article finds equations whose solution sets are the boundaries (in Cartesian coordinates) of different shapes.