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 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.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is concave downward 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 downward.

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 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.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is concave downward 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 downward.

Answer

Explanation:

Step1: Find the first derivative

The derivative of (y = f(x)=\ln(x + 5)-4) using the formula (\frac{d}{dx}\ln(u)=\frac{u'}{u}) (where (u=x + 5) and (u'=1)). So (f'(x)=\frac{1}{x+5}). Set (f'(x) = 0), (\frac{1}{x+5}=0) has no solution. Since the domain of (y=\ln(x + 5)-4) is (x>-5) and (f'(x)=\frac{1}{x + 5}>0) for all (x>-5), the function is increasing on ((-5,\infty)). A local minimum occurs where the function changes from decreasing to increasing. Since the function is always increasing, there is no local minimum.

Step2: Find the second derivative

Differentiate (f'(x)=\frac{1}{x+5}=(x + 5)^{-1}) using the power rule (\frac{d}{dx}u^n=nu^{n - 1}u'). Here (n=-1) and (u=x + 5), (u'=1). So (f''(x)=-(x + 5)^{-2}=-\frac{1}{(x + 5)^2}). Since (f''(x)=-\frac{1}{(x + 5)^2}<0) for all (x>-5) (because ((x + 5)^2>0) for (x>-5)), the function is concave - downward on ((-5,\infty)) and never concave - upward.

Answer:

B. The function (f) has no local minimum. B. The function (f) is never concave upward. A. The function (f) is concave downward on the sub - interval ((-5,\infty))