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Dr. Sachin Kaushal, Lovely Professional University Unit 13: Maxima and Minima

Unit 13: Maxima and Minima Notes

CONTENTS
Objectives
Introduction

13.1 The Extreme-value Theorem
13.1.1 First Derivative Criterion for Local Extrema
13.1.2 Second Derivative Criterion for Local Extrema

13.2 Points of Inflexion
13.2.1 N Derivative Criterion for Maxima, Minima and Point of Inflexion
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13.3 Summary
13.4 Keywords
13.5 Self Assessment

13.6 Review Questions
13.7 Further Readings

Objectives

After studying this unit, you will be able to:

Discuss the Extreme-value Theorem
Explain the points of Inflexion

Introduction

We know that the value of a function is different at different points in its domain. When the
function is monotonic, the functional values are either continuously increasing or decreasing. If
the function is not monotonic, the functional values may increase (decrease) over a certain
subset of the domain and then decrease (increase). This behaviour may be repetitive also.

13.1 The Extreme-value Theorem

If a function f(x) is continuous at every point of a closed interval I, then f(x) assumes
both an absolute maximum value M and an absolute minimum value m some where in the
interval I.
This theorem implies that there always exist two values x and x in I such that f(x ) = m, f(x ) = M
1 2 1 2
and m f(x) M for other values of x in the interval I.

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