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

for the function below, find (a) ( f^{prime}(x) ). (b) the partition numbers for ( f^{prime} ), (c) the critical numbers of ( f ).\n( f(x)=\frac{1}{x + 8} )\n(a) ( f^{prime}(x)=\frac{-1}{(x + 8)^{2}} )\n(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\na. the partition number(s) is/are ( x=-8 ). (use a comma to separate answers as needed)\nb. there are no partition numbers.\n(c) find the critical numbers of ( f ). select the correct choice below and, if necessary, fill in the answer box to complete your choice\na. the critical number(s) is/are ( x= ) (use a comma to separate answers as needed.)\nb. there are no critical numbers.

for the function below, find (a) ( f^{prime}(x) ). (b) the partition numbers for ( f^{prime} ), (c) the critical numbers of ( f ).\n( f(x)=\frac{1}{x + 8} )\n(a) ( f^{prime}(x)=\frac{-1}{(x + 8)^{2}} )\n(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\na. the partition number(s) is/are ( x=-8 ). (use a comma to separate answers as needed)\nb. there are no partition numbers.\n(c) find the critical numbers of ( f ). select the correct choice below and, if necessary, fill in the answer box to complete your choice\na. the critical number(s) is/are ( x= ) (use a comma to separate answers as needed.)\nb. there are no critical numbers.

Answer

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 either (f^{\prime}(x)=0) or (f^{\prime}(x)) is undefined. We know that (f^{\prime}(x)=\frac{-1}{(x + 8)^{2}}).

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

Set (f^{\prime}(x)=0), so (\frac{-1}{(x + 8)^{2}}=0). Since (- 1\neq0) for all real (x), the equation (\frac{-1}{(x + 8)^{2}}=0) has no solution.

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

The function (f^{\prime}(x)=\frac{-1}{(x + 8)^{2}}) is undefined when the denominator ((x + 8)^{2}=0). Solving ((x + 8)^{2}=0) gives (x=-8). But (x=-8) is not in the domain of (f(x)=\frac{1}{x + 8}) (because when (x=-8), (f(x)) has a division - by - zero error, (f(-8)=\frac{1}{-8 + 8}) is undefined).

Answer:

B. There are no critical numbers.