Question

Is `f(x)=(-5x^3+x^2-3x-11)/(x-1)` increasing or decreasing at `x=2`?

Answer 1

The function is increasing at `x=2`

Explanation:

If a function is increasing its gradient is positive and if it is decreasing its gradient is negative. Differentiate using the quotient rule to find the gradient and find the value at `x=2`
Quotient rule: If `f(x) = g(x)/h(x)` then `f'(x) = (g(x)h'(x) - g'(x)h(x))/(h^2(x))`
`= ((-5x^3+x^2-3x-11)(1) - (x-1)(-15x^2 +2x -3))/(x-1)^2`
`=(-5x^3+x^2-3x-11 -(-15x^3 +2x^2-3x +15x^2-2x +3))/(x-1)^2`
`=(10x^3-16x^2+2x-14)/(x-1)^2`

`f'(2) = (80 -64 +8 -14)/1 =88-78 = 10`
The function is increasing at `x=2`