summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y =…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=ln (x + 5)-4 )\na. the function ( f ) has a local maximum at ( x = )\n(type an exact answer. use a comma to separate answers as needed.)\nb. the function ( f ) has no local maximum.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) has a local minimum at ( x = )\n(type an exact answer. use a comma to separate answers as needed.)\nb. the function ( f ) has no local minimum.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is concave upward on the subinterval(s)\n(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed)\nb. the function ( f ) is never concave upward.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=ln (x + 5)-4 )\na. the function ( f ) has a local maximum at ( x = )\n(type an exact answer. use a comma to separate answers as needed.)\nb. the function ( f ) has no local maximum.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) has a local minimum at ( x = )\n(type an exact answer. use a comma to separate answers as needed.)\nb. the function ( f ) has no local minimum.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is concave upward on the subinterval(s)\n(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed)\nb. the function ( f ) is never concave upward.

Answer

Explanation:

Step1: Find the domain

For (y = \ln(x + 5)-4), the argument of the logarithm must be positive. So (x+5>0), which gives (x>- 5). The domain is ((-5,\infty)).

Step2: Find the first - derivative

Using the chain rule, if (y=\ln(u)-4) where (u = x + 5), then (y^\prime=\frac{dy}{du}\cdot\frac{du}{dx}). Since (\frac{d}{du}(\ln(u))=\frac{1}{u}) and (\frac{d}{dx}(x + 5)=1), we have (y^\prime=\frac{1}{x + 5}). Set (y^\prime = 0), (\frac{1}{x+5}=0) has no solution. So there are no critical points for local maxima or minima.

Step3: Find the second - derivative

Differentiate (y^\prime=\frac{1}{x + 5}=(x + 5)^{-1}) using the power rule ((u^n)^\prime=nu^{n - 1}u^\prime). Here (n=-1) and (u=x + 5), (u^\prime = 1). So (y^{\prime\prime}=-(x + 5)^{-2}=-\frac{1}{(x + 5)^2}). Since (y^{\prime\prime}<0) for all (x\in(-5,\infty)) (because ((x + 5)^2>0) for (x>-5)), the function is concave down on ((-5,\infty)) and never concave up.

Answer:

For local maximum: B. The function (f) has no local maximum. For local minimum: B. The function (f) has no local minimum. For concavity: B. The function (f) is never concave upward.