Question

Is `f(x)=(-2x^3+4x^2-x-2)/(x+3)` increasing or decreasing at `x=-2`?

Answer 1

decreasing at x = -2

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) `-2x^3+4x^2-x-2 rArr g'(x) = -6x^2+8x-1`

h(x) = x + 3 → h'(x) = 1
`"----------------------------------------------------------"`
Substitute these values into f'(x)

f'(x) `=((x+3)(-6x^2+8x-1)-(-2x^3+4x^2-x-2).1)/(x+3)^2`

and f'(-2)`=(1.(-24-16-1)-(16+16+2-2))/1`

= (-41-32) =-73

Since f'(-2) < 0 then f(x) is decreasing at x = -2
graph{(-2x^3+4x^2-x-2)/(x+3) [-10, 10, -5, 5]}