### 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

#### Contents

1 Maxima and minima2 Distinguishing between local maxima and minima

3 Engineering Example 1

3.1 Water wheel efficiency

4 Engineering Example 2

4.1 Refraction

5 Engineering Example 3

5.1 Fluid power transmission

6 Engineering Example 4

6.1 Crank used to drive a piston