What are the critical values, if any, of #f(x)= x^3/(x+4)+x^2/(x+1)-x/(x-2)#?

1 Answer
Costas C.
Mar 28, 2016

Points where #f'(x)=0#

#x=-4#

#x=-1#

#x=2#

Undefined points

#x=-6.0572#

#x=-1.48239#

#x=-0.168921#

Explanation:

If you take the derivative of the function, you will end up with:

#f'(x)=(2x^3+12x^2)/(x+4)^2+(x^2+2x)/(x+1)^2+2/(x-2)^2#

While this derivative could be zero, this function is too hard to solve without computer aid. However, the the undefined points are those that nulify a fraction. Therefore three critical points are:

#x=-4#

#x=-1#

#x=2#

By use of Wolfram I got the answers:

#x=-6.0572#

#x=-1.48239#

#x=-0.168921#

And here is the graph to show you just how difficult this is to solve:

graph{(2x^3+12x^2)/(x+4)^2+(x^2+2x)/(x+1)^2+2/(x-2)^2 [-28.86, 28.85, -14.43, 14.44]}

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