Cambridge International AS and A Level Mathematics Pure Mathematics 1

Co-ordinate geometry

P1^

2

There are some polynomials for which not all the stationary points materialise, as in

the case of y = x^4 − 4 x^3 + 5 x^2 (whose curve is shown in figure 2.27). To be accurate,

you say that the curve of a polynomial of order n has atmost n − 1 stationary points.

x

y y = x (^4) – 4x (^3) + 5x 2
–1 O 1 2 3
4
8
12
16
Figure 2.27
–1 x
y y = x (^3) – x
O 1
–1 x
y y = x^4 – x^2
O 1
x
y
y = –2x^3 + 4x^2 – 2x + 4
O 2
4
x
y y = – x^4 + 5x^2 – 4
–2 –1 O 1 2
–4
Figure 2.26
A cubic (order 3)
with two stationary
points.
A quartic (order 4)
with three turning
points.
x
y y = x 2
O
x
y
y = – x^2 + 4x
O 4
a maximum point
A quadratic
(order 2) with one
stationary point.
a minimum
point