Cambridge International AS and A Level Mathematics Pure Mathematics 1

Differentiation

P1^

5

x

y

y = –x^4

O

Figure 5.31

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
You can see that for all three of these functions both d
d
y
x
and d
d
2
2
y
x
are zero at the
origin.
Consequently, if both d
d
y
x
and d
d
2
2
y
x
are zero at a point, you still need to check the
values of d
d
y
x
either side of the point in order to determine its nature.
EXERCISE 5F   1  For each of the following functions, find d
d
y
x
and d
d
2
2
y
x

.

(i) y = x^3 (ii) y = x^5 (iii) y = 4 x^2

(iv) y = x–2 (v) y = x

(^32)
(vi) y = x^4 − (^23)
x
2  Find any stationary points on the curves of the following functions and
identify their nature.
(i) y = x^2 + 2 x + 4 (ii) y = 6 x − x^2
(iii) y = x^3 − 3 x (iv) y = 4 x^5 − 5 x^4
(v) y = x^4 + x^3 − 2 x^2 − 3 x + 1 (vi) y = x + 1
x
(vii) y = 16 x + (^12)
x
(^) (viii) y = x (^3) + 12
x
(ix) y = 6 x − x
(^32)
3  You are given that y = x^4 − 8 x^2.
(i) Find d
d
y
x

.

(ii) Find

d

d

2

2

y

x.

(iii) Find any stationary points and identify their nature.

(iv) Hence sketch the curve.