Find the intervals in which the following functions are increasing or decreasing.
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)
(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, ∞)