Answer :

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


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