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,

⇒

⇒ f’(x) = x^{3} + 2x^{2} – 5x – 6

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

⇒ f’(x) = 0

⇒ x^{3} + 2x^{2} – 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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