Question

Is `f(x)=(-3x^3+15x^2-2x+1)/(x-4)` increasing or decreasing at `x=2`?

Answer 1

decreasing at x = 2

Explanation:

To determine if a function f(x) is increasing/decreasing at x = a, 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

differentiate f(x) 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)=-3x^3+15x^2-2x+1`

`rArrg'(x)=-9x^2+30x-2`

and `h(x)=x-4rArrh'(x)=1`
`"---------------------------------------------------------------"`
Substitute these values into f'(x)

`f'(x)=((x-4)(-9x^2+30x-2)-(-3x^3+15x^2-2x+1).1)/(x-4)^2`

`rArrf'(2)=((-2)(22)-(33))/(-2)^2=-77/4`

Since f'(2) < 0 , then f(x) is decreasing at x = 2
graph{(-3x^3+15x^2-2x+1)/(x-4) [-29.65, 29.67, -14.86, 14.8]}