Question

Is `f(x)=(-3x^2-x+2)/(x^2+x)` increasing or decreasing at `x=1`?

Answer 1

decreasing at x = 1

Explanation:

To determine if a function is increasing/decreasing at x = a , we evaluate f'(a)

• If f'(a) > 0 , then f(x) is increasing at x = a

• If f'(a) < 0 , then f(x) is decreasing at x = a

differentiate f(x) using the `color(blue)" quotient rule "`

If f(x)`=g(x)/(h(x)) "then" f'(x) = (h(x).g'(x) - g(x).h'(x)) /(h(x))^2`
`"----------------------------------------------------------------"`

g(x)`= -3x^2-x+2 rArr g'(x) = -6x - 1`

and h(x) `=x^2 + x rArr h'(x) = 2x + 1`
`"---------------------------------------------------------------"`
Substitute these values into f'(x)

`f'(x) = ((x^2+x)(-6x-1) - (-3x^2-x+2)(2x+1))/(x^2+x)^2`

and f'(1) `= (2.(-7) - (-2).(3))/2^2 = (-8)/4 = -2`

Since f'(1) < 0 , then f(x) is decreasing at x = 1
graph{(-3x^2-x+2)/(x^2+x) [-10, 10, -5, 5]}