Question

Is `f(x)=(12x^2-16x-12)/(x+2)` increasing or decreasing at `x=5`?

Answer 1

`"increasing at "x=5`

Explanation:

`"to determine if f(x) is increasing at x = a evaluate"`
`f'(x)" 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)=12x^2-16x-12rArrg'(x)=24x-16`

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

`rArrf'(x)=((x+2)(24x-16)-(12x^2-16x-12))/(x+2)^2`

`rArrf'(5)=(728-208)/49>0`

`rArrf(x)" is increasing at x = 5"`