Find the intervals in which the function f(x) =x^{3}-6x^{2}+9x+15 is increasing or decreasing

Given : f(x) = x^{3} - 6x^{2} +9x +15

To find: the interval in which f(x) is increasing or decreasing

Now

f(x) = x^{3} - 6x^{2} +9x +15

f '(x) = 3x^{2}- 12x + 9

= 3 (x^{2} - 4x + 3 )

for f(x) to increase

f `(x) > 0

⇒ 3 (x^{2} - 4x + 3 ) > 0

⇒ x^{2} - 4x + 3 > 0

⇒ (x - 3) ( x - 1) > 0

⇒ In the interval -infinity < x < 1 and (union) 3 < x < +infinity f(x) is increasing and in the interval 1 < x < 3 , f(x) is decreasing Answer