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

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 \(1
eq0\) 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.