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 choic\na. the partition number(s) is/are ( x= ) (use a comma to separate answers as needed)\nb. there are no partition 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 choic\na. the partition number(s) is/are ( x= ) (use a comma to separate answers as needed)\nb. there are no partition numbers.

Answer

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{-1}{(x + 8)^{2}}). Set (f^{\prime}(x)=0): (\frac{-1}{(x + 8)^{2}}=0). Since (-1\neq0) for all (x), there is no solution for (f^{\prime}(x) = 0). Find where (f^{\prime}(x)) is undefined. The function (y = f^{\prime}(x)) is a rational function. A rational function (\frac{a}{b}) ((a=-1), (b=(x + 8)^{2})) is undefined when (b=(x + 8)^{2}=0). Solve ((x + 8)^{2}=0). Using the property that if (u^{2}=0) then (u = 0) (here (u=x + 8)), we get (x+8=0), so (x=-8).

Answer:

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