Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY CALCULUS

Q

A

B

C

f(x)

x

S

Figure 2.2 A graph of a function,f(x), showing how differentiation corre-

sponds to finding the gradient of the function at a particular point. PointsB,

QandSare stationary points (see text).

Find the third derivative of the functionf(x)=x^3 sinx.

Using (2.14) we immediately find

f′′′(x)=6sinx+3(6x)cosx+3(3x^2 )(−sinx)+x^3 (−cosx)

=3(2− 3 x^2 )sinx+x(18−x^2 )cosx.

2.1.8 Special points of a function

We have interpreted the derivative of a function as the gradient of the function at

the relevant point (figure 2.1). If the gradient is zero for some particular value of

xthen the function is said to have astationary pointthere. Clearly, in graphical

terms, this corresponds to a horizontal tangent to the graph.

Stationary points may be divided into three categories and an example of each

is shown in figure 2.2. PointBis said to be aminimumsince the functionincreases

in value in both directions away from it. PointQis said to be amaximumsince

the functiondecreasesin both directions away from it. Note thatBis not the

overall minimum value of the function andQis not the overall maximum; rather,

they are a local minimum and a local maximum. Maxima and minima are known

collectively asturning points.

The third type of stationary point is thestationary point of inflection,S.In

this case the function falls in the positivex-direction and rises in the negative

x-direction so thatSis neither a maximum nor a minimum. Nevertheless, the

gradient of the function is zero atS, i.e. the graph of the function is flat there,

and this justifies our calling it a stationary point. Of course, a point at which the