Understanding Engineering Mathematics

globalminimum of the function. Similarly for a maximum. Indeed, for a function such

asy=x+

1

x

the local minimum actually has a higher value than the local maximum

(➤297).
While the graphical representation of the behaviour of functions is very suggestive,
we need a means of distinguishing stationary points and points of inflection that depends
only on the derivatives of the function in question. The graphical form provides a hint as
to how this might work. As an example, consider a minimum point at sayx=x 0 (see
Figure 10.3).

x

x 0

y

0

y = f(x)

f(x

x increases) decreases as

f(x

) increases as

x increases

Figure 10.3Behaviour at a minimum.

To the left ofx 0 ,i.e.forx<x 0 ,y=f(x)is decreasing asxincreases, andf′(x) <0.
Forx>x 0 ,yincreases asxincreases and sof′(x) >0. And of course atx=x 0 we have
f′(x)=f′(x 0 )=0. So, near to a minimum point we can summarise the situation as in
the table below:

x<x 0 x=x 0 x>x 0

f′(x) < 0 f′(x)= 0 f′(x) > 0

This characterisation no longer relies on the graphical representation – it depends only on
the values of the derivative at different points. Rather than use this table as a means of
verifying a minimum point, we look at the implications it has for the second derivative of
the function. In this case we see that asxpasses throughx 0 ,f′(x)is steadilyincreasing.
That is

d

dx

(

dy

dx

)

=

d^2 y

dx^2

=f′′(x) > 0

So aminimum pointx=x 0 on the curvey=f(x)is characterised by

f′(x 0 )=0andf′′(x 0 )> 0

Similarly, for a maximum point we obtain the table

x<x 0 x=x 0 x>x 0

f′(x) > 0 f′(x)= 0 f′(x) < 0