Question

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

Answer 1

`"f(x) is increasing at x=0"`

Explanation:

To determine if a function f(x) 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"`

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

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

`color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=(h(x).g'(x)-g(x).h'(x))/(h(x))^2)color(white)(2/2)|)))`

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

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

`rArrf'(x)`

`=((4x-2)(-3x^2+2x-3)-(-x^3+x^2-3x-4).4)/(4x-2)^2`

`rArrf'(0)=((-2)(-3)-(-4)(4))/(-2)^2`

`=22/4=11/2`

Since f'(0) > 0 , then f(x) is increasing at x = 0
graph{(-x^3+x^2-3x-4)/(4x-2) [-20, 20, -10, 10]}