second derivative test: problem 1\n(1 point)\nconsider the function ( f(x)=5x + 8x^{-1} ).\nnote that this…

second derivative test: problem 1\n(1 point)\nconsider the function ( f(x)=5x + 8x^{-1} ).\nnote that this function has no inflection points, but ( f(x) ) is undefined at ( x = b ) where\n( b=)\nfor each of the following intervals, tell whether ( f(x) ) is concave up (type in cu) or concave down (type in cd).\n( (-infty,b):)\n( (b,infty):)\nnote: you can earn partial credit on this problem.

second derivative test: problem 1\n(1 point)\nconsider the function ( f(x)=5x + 8x^{-1} ).\nnote that this function has no inflection points, but ( f(x) ) is undefined at ( x = b ) where\n( b=)\nfor each of the following intervals, tell whether ( f(x) ) is concave up (type in cu) or concave down (type in cd).\n( (-infty,b):)\n( (b,infty):)\nnote: you can earn partial credit on this problem.

Answer

Explanation:

Step1: Find the second - derivative of (f(x))

First, find the first - derivative using the power rule ((x^n)^\prime=nx^{n - 1}). For (f(x)=5x + 8x^{-1}), (f^\prime(x)=5-8x^{-2}=5-\frac{8}{x^{2}}). Then find the second - derivative: (f^{\prime\prime}(x)=\frac{16}{x^{3}}).

Step2: Find the value of (B)

The second - derivative (f^{\prime\prime}(x)=\frac{16}{x^{3}}) is undefined when (x = 0), so (B = 0).

Step3: Determine the concavity on ((-\infty,0))

Let (x=-1) (a test point in the interval ((-\infty,0))). Then (f^{\prime\prime}(-1)=\frac{16}{(-1)^{3}}=-16<0). So (f(x)) is concave down (CD) on ((-\infty,0)).

Step4: Determine the concavity on ((0,\infty))

Let (x = 1) (a test point in the interval ((0,\infty))). Then (f^{\prime\prime}(1)=\frac{16}{(1)^{3}}=16>0). So (f(x)) is concave up (CU) on ((0,\infty)).

Answer:

(B = 0); ((-\infty,B)): CD; ((B,\infty)): CU