Question

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

Answer 1

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]}