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

1 Answer
Alan P.
Dec 7, 2017

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

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