Cambridge International AS and A Level Mathematics Pure Mathematics 1

Differentiation

P1^

5

The graph of this function is shown in figure 5.20.

Increasing and decreasing functions

When the gradient is positive, the function is described as an increasing function.

Similarly, when the gradient is negative, it is a decreasing function. These terms

are often used for functions that are increasing or decreasing for all values of x.

EXAMPLE 5.12 Show that y = x^3 + x is an increasing function.

SOLUTION

y = x^3 + x ⇒ d

d

y

x

= 3 x^2 + 1.

Since x^2  0 for all real values of x,

d

d

y

x

 1

⇒ y = x^3 + x is an increasing function.

Figure 5.21 shows its graph.

y

–0.5 0 0.5 1 t

0.875

1

–1

Figure 5.20

Figure 5.21

y

± ±   





3

±

±

±3