Question

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

Answer 1

increasing at x = 0

Explanation:

To determine if a function is increasing/decreasing at x = a ,consider the following.

• 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"`

`f(x)=(g(x))/(h(x))" then "f'(x)=(h(x).g'(x)-g(x).h'(x))/(h(x))^2`
`"--------------------------------------------------------------------"`
`g(x)=x^3-4x^2+3x-4rArrg'(x)=3x^2-8x+3`

`h(x)=4x-2rArrh'(x)=4`
`"-------------------------------------------------------------------"`
Substitute these values into f'(x)

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

and `f'(0)=((-2)(+3)-(-4).4)/(-2)^2=5/2`

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