
2-36 Calculus – Differentiation and Integration
Example
Consider the function
f x
x x
x
x x
( )
if
if
if
.=
≤ ≤
< ≤
− >
0 2
1 2 5
1 5
Clearly f is increasing because
1 2 1 1 2 2
2 5 3 2 5 1 1 3
5 7 5 4 6
< ⇒ = ≤ =
< ⇒ = ≤ =
< ⇒ = ≤ =
f f
f f
f f
( ) ( )
. ( . ) ( )
( )Also, (( )7
However, f is not strictly increasing. The function f(x) = e
x
is strictly increasing and hence strictly
monotone.
2.5.2 Decreasing Functions
A function f is said to be a decreasing function or order-reversing over an interval [a, b] if whenever
x
1
< x
2
Þ f(x
1
) ³ f(x
2
) (Fig. 2.43).
f is strictly increasing if whenever x
1
< x
2
Þ f(x
1
) > f(x
2
).
Figure 2.43
Example
Define f t
t t
t
t