Understanding Engineering Mathematics

H.Points of particular importance include maximum and minimum values(humps and
hollows)and also points of inflection. While we do already know how to find these,
it may not in fact be necessary. Information already available may hint strongly at
certain turning values – for example a continuous curve which ends up going in the
same direction for two different values ofxmust have passed through at least one
turning value in between. Also, it is not always necessary to determine points of
inflection, especially those with a non-zero gradient, unless we want a really accurate
picture of the graph.

Solution to review question 10.1.4

(i)y= 16 x− 3 x^3

First check on thesymmetry. The function is odd and so we need

only sketch it forx≥0 and then obtain the whole curve by a rotation

of 180°about the origin.

Now find thegateways. The curve crosses they-axis(x= 0 )when

y=0, i.e. it passes through the origin. It crosses thex-axis at

y= 16 x− 3 x^3 = 0

i.e. x= 0 , ±

4

√

3

There are norestrictionson the domain – the function, being a poly-

nomial, exists for all values ofx. It is continuous for all values ofx

too. There are noasymptotes. Near the origin thex^3 term is negli-

gible, andy 16 x, so it looks like a very steep straight line with

positive slope. For very large values the curve behaves likey− 3 x^3 ,

and clearly asx→∞,y→−∞.

This discussion alone is enough to tell us that there is likely to be at

least one maximum point between 0 and

4

√

3

on the positivex-axis.

Figure 10.6 illustrates this deduction.

0 x

y

y ∼−^ − 3 x^3

y ∼−^16 x

Figure 10.6Going to extremes.