Increasing and decreasing functions
Before learning the concept of
maxima and minima, we will study the nature of the curve of a given function
using derivative.
A function f(x) is said to be
increasing function in the interval [a,b] if
x1 < x2 ⇒ f
(x 1 )≤ f (x 2 ) for all x 1 , x2 ∈ |a , ]
A function f(x) is said to be
strictly increasing in [a,b] if
x1 < x2 ⇒ f
(x 1 )< f (x 2 ) for all x 1 , x2 ∈ [a ,b]
A function f(x) is said to be
decreasing function in [a, b] if
x1 < x2 ⇒ f
(x 1 )≥ f (x 2 ) for all x 1 , x2 ∈ [a ,b]
A function f(x) is said to be
strictly decreasing function in [a,b] if x
x1 < x2 ⇒ f (x 1 )> f (x 2 ) for all x 1 , x2 ∈ [a ,b]
NOTE
A function is said
to be monotonic function if it is either an increasing function or a decreasing
function.
Let f(x) be a continuous
function on [a,b] and differentiable on the open interval (a,b), then
(i) f(x)
is increasing in [a, b] if f ′ (x ) ≥ 0
(ii) f(x) is decreasing in [a,
b] if f ′ (x ) ≤ 0
(i) f(x)
is strictly increasing in (a,b) if
f ′ (x ) > 0 for every x ∈ (a ,b)
(ii) f(x)
is strictly decreasing in (a,b) if
f ′ (x ) < 0 for every x ∈ (a ,b)
(iii) f(x) is said to be a constant
function if f ′ (x ) = 0
Related Topics
Privacy Policy, Terms and Conditions, DMCA Policy and Compliant
Copyright © 2018-2024 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.