Question

consider the function ( f(x)=\frac{x - 4}{x + 2} ).
is ( x=-2 ) a critical number of ( f )?
yes
no
why or why not?
the function is not defined at ( x=-2 ).
the function is differentiable at ( x=-2 ).
the function is not differentiable at ( x=-2 ).
the function is defined at ( x=-2 ).

Explanation:

Step1: Recall the definition of critical number

A critical number of a function \(y = f(x)\) is a number \(c\) in the domain of \(f\) such that \(f^{\prime}(c)=0\) or \(f^{\prime}(c)\) does not exist.

Step2: Check the domain of \(f(x)=\frac{x - 4}{x + 2}\)

The function \(f(x)=\frac{x - 4}{x + 2}\) is a rational function. The denominator \(x + 2=0\) when \(x=-2\). So the domain of \(f(x)\) is \(\{x|x
eq - 2\}\). Since \(x =-2\) is not in the domain of \(f(x)\), it cannot be a critical number.

Answer:

No. The function is not defined at \(x=-2\).