Question

Is `f(x)=(6x^2-x-12)/(x+3)` 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 f(x) 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))^2`

`g(x)=6x^2-x-12rArrg'(x)=12x-1`

`h(x)=x+3rArrh'(x)=1`

`rArrf'(x)=((x+3)(12x-1)-(6x^2-x-12))/(x+3)^2`

`rArrf'(3)=(210-39)/36>0`

`rArrf(x)" is increasing at "x=3`