P1^
5
The(^) second
(^) derivative
157
(iv)
!^ Remember that you are looking for the value of^
d
d
2
2
y
x
at the stationary point.
Note
On occasions when it is difficult or laborious to find dd
2
2
y
x, remember that you can
always determine the nature of a stationary point by looking at the sign of ddyx
for
points just either side of it.
!^ Take care when^
d
d
2
2
y
x
= 0. Look at these three graphs to see why.
x
y y = x^3
O
Figure 5.29
y = x^3
d
d
y
x = 3x
(^2) : at (0, 0) d
d
y
x^ = 0
d
d
2
2
y
x = 6x: at (0, 0)
d
d
2
2
y
x = 0
x
y y = x^4
O
Figure 5.30
y = x^4
d
d
y
x = 4x
(^3) : at (0, 0) d
d
y
x^ = 0
d
d
2
2
y
x = 12x
(^2) : at (0, 0) d
d
2
2
y
x = 0
20
y y = 2x (^3) + 3x (^2) –12x
0 x
–7
–2 –1
Figure 5.28