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Find the intervals in which the following functions are increasing or decreasing.
f(x) = 6 – 9x – x2
Given:- Function f(x) = 6 – 9x – x2
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) = 6 – 9x – x2
⇒
⇒ f’(x) = –9 – 2x
For f(x) to be increasing, we must have
⇒ f’(x) > 0
⇒ –9 – 2x > 0
⇒ –2x > 9
⇒
⇒
⇒
Thus f(x) is increasing on interval
Again, For f(x) to be decreasing, we must have
f’(x) < 0
⇒ –9 – 2x < 0
⇒ –2x < 9
⇒
⇒
⇒
Thus f(x) is decreasing on interval