Math in Minutes: 200 Key Concepts Explained In An Instant by Paul Glendinning

Author:Paul Glendinning
Language: eng
Format: mobi
Tags: MATHEMATICS
ISBN: 9781623650094
Publisher: Quercus
Published: 2013-09-09T14:00:00+00:00

Maxima and minima

The process of finding the maximum or minimum values of a function is called optimization. A maximum of a function ƒ(x) lies at point c if ƒ(c) is greater than or equal to ƒ(x) for all other values of x. Similarly, a minimum lies at d if ƒ(d) is less than or equal to ƒ(x) for all other x. A local maximum or minimum is one where ƒ(x) is only compared for nearby values of x.

At these points, the tangent to the curve is horizontal, so the derivative is zero. This provides an easy way to determine local maxima or minima. At a point c where the derivative is zero, the linear term of the Taylor series (see page 222) disappears and

+ higher order terms

If then this is locally like a parabola, with a maximum if the second derivative is negative and a minimum if it is positive. If it may instead be a point of inflection, where the function flattens out before continuing in the same direction.

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