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 ) is increasing on the subinterval(s) ( (-5, infty) ).\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function ( f ) is never increasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is decreasing on the subinterval(s)\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function ( f ) is never decreasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\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.

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 ) is increasing on the subinterval(s) ( (-5, infty) ).\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function ( f ) is never increasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is decreasing on the subinterval(s)\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function ( f ) is never decreasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\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.

Answer

Explanation:

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

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}).

Step2: Analyze the sign of the derivative

The domain of (f(x)) is (x+5>0) or (x>-5). For (x\in(-5,\infty)), (f'(x)=\frac{1}{x + 5}>0). Since (f'(x)>0) on ((-5,\infty)), the function is increasing on ((-5,\infty)) and never decreasing.

Step3: Analyze local maxima/minima

A local maximum (or minimum) occurs where (f'(x) = 0) or (f'(x)) is undefined. (f'(x)=\frac{1}{x+5}) is never equal to (0) (because (1\neq0) for all (x) in the domain) and is undefined at (x=-5) (but (x =- 5) is not in the domain of (f(x))). So the function has no local maximum.

Answer:

A. The function (f) is increasing on the subinterval(s) ((-5,\infty)). B. The function (f) is never decreasing. B. The function (f) has no local maximum.