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= ). (use a comma to separate answers as needed.)
b. there are no partition numbers.

Explanation:

Step1: Recall the definition of partition numbers

Partition numbers of \(f^{\prime}(x)\) are the values of \(x\) where \(f^{\prime}(x) = 0\) or \(f^{\prime}(x)\) is undefined.
Given \(f^{\prime}(x)=-\frac{5}{(x + 8)^{2}}\).

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

Set \(f^{\prime}(x)=0\), so \(-\frac{5}{(x + 8)^{2}}=0\).
Since \(-5
eq0\) and \((x + 8)^{2}>0\) for all real \(x
eq - 8\), there is no solution for \(f^{\prime}(x)=0\).

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

The function \(y = f^{\prime}(x)\) 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 \(x+8 = 0\), so \(x=-8\).

Answer:

A. The partition number(s) is/are \(x=-8\)