### 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

4.1 Refraction