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) lets find critical point, we must have

⇒ f’(x) = 0

⇒

⇒

⇒ x = 0, 1

Since x > 0, therefore only check the range on the positive side of the number line.

clearly, f’(x) > 0 if 0 < x < 1

and f’(x) < 0 if x > 1

Thus, f(x) increases on (0, 1)

and f(x) is decreasing on interval x ∈ (1, ∞)

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