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 )
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function ( f ) is concave upward on the subinterval(s)
(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)
a. the function ( f ) is never concave upward.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function ( f ) is concave downward on the subinterval(s) ( (-5, infty) ).
(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)
a. the function ( f ) is never concave downward.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function ( f ) has an inflection point at ( x = )
(type an exact answer. use a comma to separate answers as needed.)
a. the function ( f ) has no inflection point.

Explanation:

Step1: Find the first derivative

The function is \(y = f(x)=\ln(x + 5)-4\).
Using the formula \(\frac{d}{dx}\ln(u)=\frac{u'}{u}\) (where \(u=x + 5\) and \(u'=1\)), we get \(f'(x)=\frac{1}{x + 5}\).

Step2: Find the second derivative

Differentiate \(f'(x)=\frac{1}{x + 5}=(x + 5)^{-1}\) using the power rule \(\frac{d}{dx}x^{n}=nx^{n-1}\).
\(f''(x)=-(x + 5)^{-2}=-\frac{1}{(x + 5)^{2}}\).

Step3: Analyze concavity

For concavity, we consider the sign of \(f''(x)\).
Since \((x + 5)^{2}>0\) for all \(x
eq - 5\) (the domain of \(f(x)\) is \(x>-5\)), then \(f''(x)=-\frac{1}{(x + 5)^{2}}<0\) for all \(x\in(-5,\infty)\).
A function \(y = f(x)\) is concave upward when \(f''(x)>0\) and concave downward when \(f''(x)<0\).
Since \(f''(x)<0\) for all \(x\) in the domain \((-5,\infty)\) of \(f(x)\), there is no interval where \(f''(x)>0\).
An inflection point occurs where \(f''(x) = 0\) or \(f''(x)\) is undefined and the concavity changes. Since \(f''(x)
eq0\) for all \(x\) in the domain of \(f(x)\) (and \(f''(x)\) is defined for all \(x>-5\)), there is no inflection point.

Answer:

• For concavity upward: B. The function \(f\) is never concave upward.
• For concavity downward: A. The function \(f\) is concave downward on the sub - interval(s) \((-5,\infty)\)
• For inflection point: B. The function \(f\) has no inflection point.