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

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Answer 1 · 2017-01-07T11:18:05

At #x=1, f(x)# is decreasing

Let's calculate (by the rule of derivatove of a division):

#f'(x)=((-3x^2-4x-1)(x^2+x)-(2x+1)(-x^3-2x^2-x+2))/((x^2+x)^2)#

Then let's substitute x=1:

#f'(1)=((-3-4-1)(1+1)-(2+1)(-1cancel(-2)-1cancel(+2)))/((1+1)^2)#

#=(-16+6)/4=-5/2<0#

Since #f'(1)<0# then f(x) is decreasing at x=1