Question

Is `f(x)=(-2x^3-x^2-5x+2)/(x+1)` increasing or decreasing at `x=0`?

Answer 1

decreasing at x = 0

Explanation:

To test if a function is increasing / decreasing at x = a , require to check the sign of 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

Require to find f'(x)

differentiate 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 - x^2- 5x + 2 rArr g'(x) =-6x^2-2x-5`

h(x) = x+1 `rArr h'(x) = 1`
`"------------------------------------------------------------------------"`
substitute these values into f'(x)

`rArr f'(x) =( (x+1).(-6x^2-2x-5) - (-2x^3-x^2-5x+2).1)/(x+1)^2`

and f'(0) = `(1.(-5) - 2.1)/1 = -7`

since f'(0) < 0 , f(x) is decreasing at x = 0

Here is the graph of f(x)
graph{(-2x^3-x^2-5x+2)/(x+1) [-10, 10, -5, 5]}