Understanding Engineering Mathematics

and so in this case the derivative isdecreasingas we increasexthroughx=x 0 and thus
amaximum pointx=x 0 is characterised by

f′(x 0 )= 0 f′′(x 0 )< 0

Now what happens iff′′(x 0 )=0? Then, it is not clear whether or not thegradient,f′(x),
is changing atx=x 0 and a more careful examination is necessary. An example of one
possibility occurs at the pointEin Figure 10.2, at which point

f′(x 0 )=f′′(x 0 )= 0

This is a particular example of apoint of inflection.BandDare other examples, but
these are clearly not stationary points – the gradient is not zero. In these casesf′′(x 0 )= 0
butf′(x 0 )=0. So what precisely characterises a point of inflection?
Look more closely at the pointBin Figure 10.2. To emphasise the point we will bend
the curve somewhat near toB, as in Figure 10.4. Takex=x 0 as the coordinate ofBhere.

x

B

x 0

Figure 10.4Detail of a point of inflection.

On either side ofBthe gradient is negative. However, on the left ofB,x<x 0 ,the
curve isbelow the tangent. We say the curve here isconcave down. To the right ofB,
x>x 0 ,thecurveisabove the tangentand we say it isconcave up. Looking at points
DandEwe see a similar change:

left of D – concave up

right of D – concave down

left of E – concave down

right of E – concave up

In general, any point where there is such a change of sense of concavity is called apoint
of inflection, meaning a change in the direction of bending of the curve. At such a point
you will see that it is the rate of change of thegradientthat is changing sign. For example:

left ofB, gradient steepens asxincreases – curve concave down

right ofB, gradient decreases asxincreases – curve concave up

So even though the gradient off(x)may not be zero at a point of inflection – i.e. we
may not have a stationary value – the rate of change of the gradient,f′′(x 0 ),atsucha
point must be zero. However, note that the conditionf′′(x 0 )=0 does not itself guarantee
a point of inflection. The Review Question 10.1.4(v) illustrates this.