Is #f(x)=(-12x^2-22x-2)/(x-4)# increasing or decreasing at #x=1#?

1 Answer
Jan 21, 2016

The function is increasing at #x=1#

Explanation:

If #(dy)/(dx)# is positive then increasing, otherwise decreasing.

Given:#" "color(brown)( y=(-12x^2-22x-2)/(x-4))#

Using #y=u/v -> (dy)/(dx)= (v(du)/(dx)-u(dv)/(dx))/(v^2)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#f'(x)=((x-4)(-24x-22)-(-12x^2-22x-2)(1))/((x-4)^2)#

At #x=1# we have:

#f'(x)=((1-4)(-24-22)-(-12-22-2)(1))/((1-4)^2)#

#f'(x)= ("positive " +" possitive")/("possitive")-> "positive solution"#

Thus as#f'(x)" "#is positive at #x=1# the function is increasing!