Question

Is `f(x)=(x^3+x^2+2x-4)/(4x-2)` increasing or decreasing at `x=0`?

Answer 1

For `x=0`,
`f’(x)=-5` or `f’(x)<0` and hence decreasing.

Explanation:

`u=x^3+x^2+2x-4`
`du=3x^2+2x+2`
`v=4x-2`, `dv=4`
Applying quotient rule,
`f’(x)=((v*du)-(u*dv))/v^2`
`f’(x)=(((4x-2)(3x^2+2x+2))-((x^3+x^2+2x-4)*4))/(4x-2)^2`
`f’(x)=(12x^3+8x^2+8x-6x^2-4x-4-4x^3-4x^2+8x-16)/(4x-2)^2`
`f’(x)=(8x3-2x^2+12x-20)/(4x-2)^2`
For `x=0`,
`f’(x)=-20/4=-5` or `f’(x)<0` and hence decreasing.