P1^
5
Maximum(^) and
(^) minimum
(^) points
147
You will see that
●●at the maximum point the gradient d
d
y
x
is zero
●●the gradient is positive to the left of the maximum and negative to the right of it.
This is true for any maximum point (see figure 5.15).
In the same way, for any minimum point (see figure 5.16):
●●●the gradient is zero at the minimum
●●the gradient goes from negative to zero to positive.
Maximum and minimum points are also known as stationary points as the
gradient function is zero and so is neither increasing nor decreasing.
EXAMPLE 5.10 Find the stationary points on the curve of y = x^3 − 3 x + 1, and sketch the curve.
SOLUTION
The gradient function for this curve is
d
d
y
x
= 3 x^2 − 3.
The x values for which d
d
y
x
= 0 are given by
3 x^2 − 3 = 0
3(x^2 − 1) = 0
3(x + 1)(x − 1) = 0
⇒ x = −1 or x = 1.
The signs of the gradient function just either side of these values tell you the
nature of each stationary point.
0
- –
Figure 5.15
0
Figure 5.16