Question

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

Explanation:

Step1: Find the domain of the function

The domain of \(y = \ln(x + 5)-4\) is determined by the argument of the natural logarithm. For \(y=\ln(u)\), \(u>0\). Here \(u=x + 5\), so \(x+5>0\) or \(x>- 5\). The domain is \((-5,\infty)\).

Step2: Find the first - derivative

Using the chain rule, if \(y=\ln(x + 5)-4\), then \(y^\prime=\frac{d}{dx}(\ln(x + 5))-\frac{d}{dx}(4)\). Since \(\frac{d}{dx}(\ln(u))=\frac{u^\prime}{u}\) (where \(u=x + 5\) and \(u^\prime = 1\)) and \(\frac{d}{dx}(4)=0\), we have \(y^\prime=\frac{1}{x + 5}\).

Step3: Analyze the sign of the first - derivative

For \(x\in(-5,\infty)\), \(x + 5>0\), so \(y^\prime=\frac{1}{x + 5}>0\) for all \(x\in(-5,\infty)\).
Since the first - derivative \(y^\prime>0\) for all \(x\) in the domain of \(y = f(x)\), the function is increasing on \((-5,\infty)\) and never decreasing.
For a local maximum or minimum, we look at the critical points. Critical points occur where \(y^\prime = 0\) or \(y^\prime\) is undefined.
\(y^\prime=\frac{1}{x + 5}\) is never equal to \(0\) (because \(1
eq0\) for any \(x\)) and is undefined at \(x=-5\) (but \(x =-5\) is not in the domain of \(y = f(x)\)).

Answer:

B. The function \(f\) is never decreasing.
B. The function \(f\) has no local maximum.
B. The function \(f\) has no local minimum.