Question

Is `f(x)=(-x^3-2x^2-12x+2)/(x-4)` increasing or decreasing at `x=3`?

Answer 1

`f(x)` is increasing at `x=3`

Explanation:

Simplifying `(-x^3-2x^2-12x+2)/(x-4)`
by synthetic division:
#{: (," | ",color(gray)(x^3),color(grey)(x^2),color(gray)(x^1),color(grey)(x^0)), (," | ",-1,-2,-12,+2), (ul(+color(white)("xx"))," | ",ul(color(white)("XX")),ul(-4),ul(-24),ul(-124)), (xx4," | ",-1,-6,-36,-126), (,,color(gray)(x^2),color(gray)(x^1),color(gray)(x^0),color(gray)(x^(-1))) :}#

`f(x)=-x^2-6x-36-126x^(-1)`

`f'(x)=2x-6color(white)("xx")+126x^(-2)`

`f'(3) = -6 -6 +126/(3^2)`
`color(white)("XXX")=12+14`
`color(white)("XXX")=+2`

Since `f'(x) > 0` at `x=3`
`f(x)` is increasing at `x=3`