Find the intervals in which the following functions are increasing or decreasing.

f(x) = 6 – 9x – x^{2}

Given:- Function f(x) = 6 – 9x – x^{2}

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 – x^{2}

⇒

⇒ 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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