Question

Is `f(x)=(-2x^3+x^2-2x-4)/(4x-2)` increasing or decreasing at `x=0`?

Answer 1

`"increasing at x = 0"`

Explanation:

`"to determine if f(x) is increasing/decreasing at x = a"`
`"differentiate and evaluate at x = a"`

`• " if "f'(x)>0" then f(x) is increasing at x = a"`

`• " if "f'(x)<0" then f(x) is decreasing at x = a"`

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

`"given "f(x)=(g(x))/(h(x))" then"`

`f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larrcolor(blue)"quotient rule"`

`g(x)=-2x^3+x^2-2x-4rArrg'(x)=-6x^2+2x-2`

`h(x)=4x-2rArrh'(x)=4`

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

`rArrf'(0)=((-2)(-2)-4(-4))/4=5>0`

`"since " f'(x)>0" then f(x) is increasing at x = 0"`
graph{(-2x^3+x^2-2x-4)/(4x-2) [-10, 10, -5, 5]}