Question

Is `f(x)=(x^3-4x^2-4x+5)/(x+2)` increasing or decreasing at `x=3`?

Answer 1

`"increasing at "x=3`

Explanation:

`"to determine if f(x) is increasing/decreasing at x = a"`
`"differentiate and evaluate at x = 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 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))^2larr" quotient rule"`

`g(x)=x^3-4x^2-4x+5rArrg'(x)=3x^2-8x-4`

`h(x)=x+2rArrh'(x)=1`

`rArrf'(x)=((x+2)(3x^2-8x-4)-(x^3-4x^2-4x+5))/(x+2)^2`

`rArrf'(3)=(5(-1)-(-16))/25=11/25`

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