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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