Question

Is `f(x)=(-12x^2-22x-2)/(x-4)` increasing or decreasing at `x=1`?

Answer 1

The function is increasing at `x=1`

Explanation:

If `(dy)/(dx)` is positive then increasing, otherwise decreasing.

Given:`" "color(brown)( y=(-12x^2-22x-2)/(x-4))`

Using `y=u/v -> (dy)/(dx)= (v(du)/(dx)-u(dv)/(dx))/(v^2)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`f'(x)=((x-4)(-24x-22)-(-12x^2-22x-2)(1))/((x-4)^2)`

At `x=1` we have:

`f'(x)=((1-4)(-24-22)-(-12-22-2)(1))/((1-4)^2)`

`f'(x)= ("positive " +" possitive")/("possitive")-> "positive solution"`

Thus as`f'(x)" "`is positive at `x=1` the function is increasing!