Question

Is `f(x)=(3x^3-5x^2+13x-11)/(x-3)` increasing or decreasing at `x=2`?

Answer 1

Increasing

Explanation:

Use the quotient rule of differentiation to find the gradient. Then if the gradient is negative the function is decreasing, and if it is positive the function is increasing.
If `f(x) = g(x)/h(x)`
`f'(x) =(g(x)h'(x) - g'(x)h(x))/(h^2(x))`

`f'(x) = ((3x^3 - 5x^2 +13x -11)(1) - (9x^2 -10x +13)(x-3))/(x-3)^2`
`=((3x^3 - 5x^2 +13x -11) - (9x^3-10x^2+13x-27x^2+30x-39))/(x-3)^2`
`=(-6x^3 + 32x^2 -30x+28)/(x-3)^2`
`=(-2(3x^3 -16x^2+15x-14))/(x-3)^2`
Substituting in `x=2` gives `(-2(24-64+30-14))/1`
`=-2(54 - 78) = -2*(-24) = 48`

The gradient (slope) of `48` is positive and the function is therefore increasing.