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Find the intervals in which the following functions are increasing or decreasing.
f(x) = x2 + 2x – 5
Given:- Function f(x) = x2 + 2x – 5
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) = x2 + 2x – 5
⇒
⇒ f’(x) = 2x + 2
For f(x) to be increasing, we must have
⇒ f’(x) > 0
⇒ 2x + 2 > 0
⇒ 2x < –2
⇒
⇒ x < –1
⇒ x ∈ (–∞,–1)
Thus f(x) is increasing on interval (–∞,–1)
Again, For f(x) to be increasing, we must have
f’(x) < 0
⇒ 2x + 2 < 0
⇒ 2x > –2
⇒
⇒ x> –1
⇒ x ∈ (–1,∞)
Thus f(x) is decreasing on interval x ∈ (–1, ∞)