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\n( f(x)=\frac{8}{x + 1} )\n\n(a) ( f^{prime}(x)= )\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.\n\na. the partition number(s) is/are ( x= ). (use a comma to separate answers as needed.)\n\nb. there are no partition numbers.\n\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.\n\na. the critical number(s) is/are ( x= ). (use a comma to separate answers as needed.)\n\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\n( f(x)=\frac{8}{x + 1} )\n\n(a) ( f^{prime}(x)= )\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.\n\na. the partition number(s) is/are ( x= ). (use a comma to separate answers as needed.)\n\nb. there are no partition numbers.\n\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.\n\na. the critical number(s) is/are ( x= ). (use a comma to separate answers as needed.)\n\nb. there are no critical numbers.

Answer

Explanation:

Step1: Find the derivative of ( f(x)=\frac{8}{x + 1}=8(x + 1)^{-1} )

Using the power rule ( (u^n)^\prime=nu^{n - 1}u^\prime ) (where ( u=x + 1), (n=-1), and (u^\prime = 1)): ( f^\prime(x)=8\times(-1)(x + 1)^{-2}\times1=-\frac{8}{(x + 1)^2} )

Step2: Find the partition numbers of ( f^\prime(x) )

Partition numbers of ( f^\prime(x) ) are the values of ( x ) where ( f^\prime(x) = 0 ) or ( f^\prime(x) ) is undefined. Set ( f^\prime(x)=0 ): ( -\frac{8}{(x + 1)^2}=0 ), no solution since the numerator is non - zero. Set the denominator of ( f^\prime(x)=(x + 1)^2 = 0 ), then ( x=-1 )

Step3: Find the critical numbers of ( f(x) )

Critical numbers of ( f(x) ) are the values of ( x ) in the domain of ( f(x) ) where ( f^\prime(x)=0 ) or ( f^\prime(x) ) is undefined. The domain of ( f(x)=\frac{8}{x + 1} ) is ( x\neq - 1 ). So there are no critical numbers (because (x=-1) is not in the domain of (f(x)))

Answer:

(A) ( f^\prime(x)=-\frac{8}{(x + 1)^2} ) (B) A. The partition number(s) is/are (x=-1) (C) B. There are no critical numbers.