Question

for the function below, find (a) ( f^{prime}(x) ), (b) the partition numbers for ( f^{prime} ), (c) the critical numbers of ( f ).

( f(x)=\frac{5}{x + 8} )

(a) ( f^{prime}(x)=-\frac{5}{(x + 8)^{2}} )

(b) find the partition numbers for ( f^{prime} ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the partition number(s) is/are ( x=-8 ). (use a comma to separate answers as needed.)
b. there are no partition numbers.

(c) find the critical numbers of ( f ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the critical number(s) is/are ( x=square ). (use a comma to separate answers as needed.)
b. there are no critical numbers.

Explanation:

Step1: Recall the definition of critical numbers

Critical numbers of a function \(y = f(x)\) are the values of \(x\) in the domain of \(f\) where \(f^{\prime}(x)=0\) or \(f^{\prime}(x)\) is undefined.
We know that \(f^{\prime}(x)=-\frac{5}{(x + 8)^{2}}\).

Step2: Check when \(f^{\prime}(x)=0\)

Set \(f^{\prime}(x)=0\), so \(-\frac{5}{(x + 8)^{2}}=0\).
Since \(- 5
eq0\) for all real \(x\), the equation \(-\frac{5}{(x + 8)^{2}}=0\) has no solution.

Step3: Check when \(f^{\prime}(x)\) is undefined

The function \(f^{\prime}(x)=-\frac{5}{(x + 8)^{2}}\) is a rational - function. A rational function \(\frac{a}{b}\) (where \(a=-5\) and \(b=(x + 8)^{2}\)) is undefined when \(b = 0\).
Set \((x + 8)^{2}=0\), then \(x+8 = 0\), which gives \(x=-8\). But \(x=-8\) is not in the domain of \(f(x)=\frac{5}{x + 8}\) (because when \(x=-8\), \(f(x)\) is undefined as the denominator is zero).

Answer:

B. There are no critical numbers.