Introduction
In this Section we analyse curves in the ‘local neighbourhood’ of a stationary point and, from this analysis, deduce necessary conditions satisfied by local maxima and local minima. Locating the maxima and minima of a function is an important task which arises often in applications of mathematics to problems in engineering and science. It is a task which can often be carried out using only a knowledge of the derivatives of the function concerned. The problem breaks into two parts
- finding the stationary points of the given functions
- distinguishing whether these stationary points are maxima, minima or, exceptionally, points of inflection.
This Section ends with maximum and minimum problems from engineering contexts.
Prerequisites
- be able to obtain first and second derivatives of simple functions
- be able to find the roots of simple equations
Learning Outcomes
- explain the difference between local and global maxima and minima
- describe how a tangent line changes near a maximum or a minimum
- locate the position of stationary points
- use knowledge of the second derivative to distinguish between maxima and minima