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Find the intervals in which the following functions are increasing or decreasing.
Given:- Function
Theorem:- Let f be a differentiable real function defined on an open interval (a,b).
(i) If f’(x) > 0 for all , then f(x) is increasing on (a, b)
(ii) If f’(x) < 0 for all , then f(x) is decreasing on (a, b)
Algorithm:-
(i) Obtain the function and put it equal to f(x)
(ii) Find f’(x)
(iii) Put f’(x) > 0 and solve this inequation.
For the value of x obtained in (ii) f(x) is increasing and for remaining points in its domain it is decreasing.
Here we have,
⇒
⇒
⇒
⇒
⇒
⇒
For f(x) to be increasing, we must have
⇒ f’(x) > 0
⇒
⇒ (x – 2) > 0
⇒ 2 < x < ∞
⇒ x ∈ (2, ∞)
Thus f(x) is increasing on interval (2, ∞)
Again, For f(x) to be decreasing, we must have
f’(x) < 0
⇒
⇒ (x – 2) < 0
⇒ –∞ < x < 2
⇒ x ∈ (–∞, 2)
Thus f(x) is decreasing on interval (–∞, 2)