The given function f(x) = (x + 6)/(x² - 9x + 18) is
(a) increasing on: (-6 - 6√3, 3), (3, -6 + 6√3)
(b) decreasing on: (-∞, -6-6√3), (-6+6√3,6), (6, ∞)
To find the increase in a function, the function f(x) is to be derivated and the derivative must be greater than 0. I.e,
f'(x) > 0
To find the decrease in a function, the function f(x) is to be derivated and the derivative must be less than 0. i.e.,
f'(x) < 0
The given function is f(x) = (x + 6)/(x² - 9x + 18).
Its graph is shown below.
To find the increase or decrease in the function, we need to derivate it.
So,
f'(x) = d/dx{(x + 6)/(x² - 9x + 18)}
Applying d/dx(u/v) = (vu'-uv')/v²
So, on derivating the given function, we get
f'(x) = (x²+12x+72)/(x²-9x+18)²
= (x - (-6 + 6√3)) (x- (-6 - 6√3))/(x - 3)²(x - 6)²
For f'(x) > 0
The intervals that satisfy the given function for increasing are
(-6 - 6√3, 3), (3, -6 + 6√3)
For f'(x) < 0
The intervals that satisfy the given function for decreasing are
(-∞, -6-6√3), (-6+6√3,6), (6, ∞)
So, from the graph and the roots obtained, we can write,
The function f(x) decreases on: (-∞, -6-6√3), (-6+6√3,6), (6, ∞) and increases on (-6 - 6√3, 3), (3, -6 + 6√3).
Of the options none of them are correct.
Learn more about increase or decrease on a function here:
https://brainly.com/question/1503051
#SPJ1
