critical numbers of f'(x) = [(x+1)^2 (4-3x)^3]/(x^2+1)^2

[Becky4paws]

In this problem, the critical numbers aren't 'obvious'. How would I go about finding the critical numbers of this derivative:

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

 

[pka]

If I were you, I would rewrite this as:
\(\displaystyle \left( {x + 1} \right)^2 \left( {4 - 3x} \right)^3 \left( {x^2 + 1} \right)^{ - 2}\)

Then apply the following:
\(\displaystyle \begin{array}{l}
j(x) = f(x)g(x)h(x) \\
j'(x) = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) \\
\end{array}\)