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# Find the interval

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, ∞)

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