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2.19 INCREASING AND DECREASING
FUNCTIONS

Let f be defined on an interval (a, b) and let x1, x2 ∈(a,b) . Then

(i) f is increasing on the interval (a, b) if f(x2) > f(x1) whenever x2 > x1
(ii) f is decreasing on the interval (a, b) if f(x2) < f(x1) whenever x2 > x1

We see that a differentiable function f is increasing on (a,b) if tangent lines to its graph

at all points (x, f(x)) where xd(a, b) have positive slopes, that is,
f ’ (x) > 0 for all x such that a < x < b

and f is decreasing on (a, b) if tangent lines to its graph at all points ( x, f ( x)) where

x ∈(a,b) , have negative slopes, that is, f '( x) < 0 for all x such that a < x < b

Now we state the above observation in the following theorem.

Theorem: version: 1.1
Let f be a differentiable function on the open interval (a,b). Then

(i) f is increasing on (a,b) if f ' ( x) > 0 for each x ∈(a,b)
(ii) f is decreasing on (a,b) if f ' ( x) < 0 for each x ∈(a,b)
Let f ( x) = x2 , then
f ( x2 ) - f ( x1 ) = x22 - x12 = ( x2 - x1 )( x2 + x1 )
If x1,x2 ∈(-∞, 0) and x2 > x1 ,, then

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