Q. 1 T5.0( 2 Votes )

Find the intervals in which the following functions are increasing or decreasing.


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,




f’(x) = x3 + 2x2 – 5x – 6


For f(x) lets find critical point, we must have


f’(x) = 0


x3 + 2x2 – 5x – 6 = 0


(x+1)(x – 2)(x + 3) = 0


x = –1, 2 , –3





clearly, f’(x) > 0 if –3 < x < –1 and x > 2


and f’(x) < 0 if x < –3 and –3 < x < –1


Thus, f(x) increases on (–3, –1) (2, ∞)


and f(x) is decreasing on interval (∞, –3) (–1, 2)



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