Find the intervals in which the following functions are increasing or decreasing.
f(x) = 10 – 6x – 2x2
Given:- Function f(x) = 10 – 6x – 2x2
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) = 10 – 6x – 2x2
⇒
⇒ f’(x) = –6 – 4x
For f(x) to be increasing, we must have
⇒ f’(x) > 0
⇒ –6 –4x > 0
⇒ –4x > 6
⇒
⇒
⇒
Thus f(x) is increasing on the interval
Again, For f(x) to be increasing, we must have
f’(x) < 0
⇒ –6 –4x < 0
⇒ –4x < 6
⇒
⇒
⇒
Thus f(x) is decreasing on interval