Understanding Engineering Mathematics

so the equation of the normal is

y− 1 =−^13 (x− 2 )

or

x+ 3 y− 5 = 0

10.2.3 Stationary points and points of inflection

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Since the derivative describes rate of change, or the slope, of a curve it can tell us a
great deal about the shape of a curve, and the corresponding behaviour of the function.
Figure 10.2 shows the range of possibilities one can meet.

x

y

0

y = f(x)

ABC D E

Figure 10.2Stationary points and points of inflection.

In all cases we assume that the function is continuous and smooth (the graph has no
breaks and no sharp points). At pointsA,C,E,wheredy/dxis zero,yis not actually
changing at all asxvaries – the tangent to the curve is thenparallel to thex-axisat
such points as shown. These are calledstationary points. So, at a stationary point of the
functiony=f(x)we have

dy

dx

=f′(x)= 0

The value off(x)at a stationary point is called astationary valueoff(x). Note that we
will usually use the ‘dash’ notation for derivatives in the rest of the book – it saves space!
There are a number of different types of stationary points illustrated in Figure 10.2,
displaying the different possibilities –Cis aminimumpoint, andAis amaximum.Eis
a stationary point where the tangent crosses the curve – this is an example of an important
point called apointofinflection.BandDare also points of inflection of a different
kind – the tangent crosses the curve, but is not horizontal – see below. The maximum and
minimum points are calledturning points, since the curve turns at such points and changes
direction. As emphasised above, differentiation is alocalprocedure. So, for example any
minimum identified by differentiation is only alocalminimum, not necessarily an overall