APFIVE Critical Numbers of a Rational Function
Which of the following statements is true regarding the critical numbers of the function $$f(x)=\frac{x}{x+1}$$ on the domain $$(-\infty,\infty)$$?
A
There are infinitely many critical numbers because the derivative is a constant positive value, indicating no change in the behavior of $$f(x)$$.
B
The function has no critical numbers because its derivative $$f'(x)=\frac{1}{(x+1)^2}$$ is always positive for $$x\neq -1$$, and $$x=-1$$ is not in the domain.
C
Critical numbers occur where the derivative is undefined; hence, $$x=-1$$ is the unique critical number of $$f(x)$$.
D
The only critical number is $$x=-1$$, where the derivative is zero.
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