Answer :

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)


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