Question

Is `f(x)=(-5x^3-x^2-3x-11)/(x-3)` increasing or decreasing at `x=2`?

Answer 1

`"increasing at "x=2`

Explanation:

To determine if 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"`

`"Given "f(x)=(g(x))/(h(x))" then"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2)color(white)(2/2)|)))`

`"here "g(x)=-5x^3-x^2-3x-11`

`rArrg'(x)=-15x^2-2x-3`

`"and "h(x)=x-3rArrh'(x)=1`

`rArrf'(x)=((x-3)(-15x^2-2x-3)-(-5x^3-x^2-3x-11)(1))/(x-3)^2`

`rArrf'(2)=((-1)(-67)-(-61))/1=128`

Since f'(2) > 0, then f(x) is increasing at x = 2
graph{(-5x^3-x^2-3x-11)/(x-3) [-10, 10, -5, 5]}