Summarize the pertinent information oblaned by applying tho graphing stralegy and skatch the grap' of $$ y=f(x) $$. $$ T(x)=\ln (x+8)-1 $$ Select the correct choice below and, if necessary, fill in the answer box lo complate your choice. A. Tho function $$ t $$ has a local maximum at $$ x= $$ $$ \square $$ (T)pe an exact answer Use a conma lo separate answers as noedod.) B. The lunction thas no local maximum. Select the corred choice below and, if necessary, fill in the answer box to complete your choice. A. The funcion I has a local minimum at $$ \mathrm{x}= $$ $$ \square $$ (Type an exact answer. Use a comma to separate answers as needed.) B. The function f has no local minimum. Select the conect choice below and, if necessary, fill in the answer box to complete your chrice. A. The function $$ f $$ is concave upward on the subintervalis) $$ \square $$ $$ \square $$. (Trpo your answer in inlerval notation. Type an oxact answor. Use a comma lo separale answers as needed.) B. The function it never concave upward.

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### Domain and Key Features

- The function is 
  $$
  T(x)=\ln(x+8)-1.
  $$
- The domain is 
  $$
  x > -8,
  $$
  because $$ \ln(x+8) $$ is defined only for $$ x+8 > 0 $$.
- There is a vertical asymptote at 
  $$
  x=-8.
  $$
- As $$ x\to -8^+ $$, the function tends to $$ -\infty $$, and as $$ x\to \infty $$, $$ T(x)\to \infty $$ (increasing slowly).

### Derivative Analysis

- The first derivative is
  $$
  T'(x)=\frac{1}{x+8}.
  $$
- Since $$ x+8>0 $$ for all $$ x $$ in the domain, we have 
  $$
  T'(x)>0 \quad \text{for } x>-8.
  $$
- This implies the function is strictly increasing, so it has **no local maximum** nor **local minimum** in its domain.

### Second Derivative and Concavity

- The second derivative is
  $$
  T''(x)=-\frac{1}{(x+8)^2}.
  $$
- Since $$ (x+8)^2>0 $$ for $$ x>-8 $$, it follows that 
  $$
  T''(x)<0 \quad \text{for all } x>-8.
  $$
- This indicates the function is concave downward everywhere in its domain and is never concave upward.

### Answer to the Multiple‐Choice Questions

1. **Local Maximum:**  
   B. The function $$ t $$ has no local maximum.

2. **Local Minimum:**  
   B. The function $$ f $$ has no local minimum.

3. **Concavity:**  
   B. The function $$ f $$ is never concave upward.

### Sketch of the Graph

- **Starting behavior:** For $$ x $$ near $$-8$$ (from the right), $$ T(x) $$ plummets down toward $$ -\infty $$ because of the vertical asymptote at $$ x=-8 $$.
- **Increasing:** As $$ x $$ increases, the function steadily and slowly increases.
- **Shape:** The graph resembles the typical logarithmic curve that has been shifted $$ 8 $$ units to the left and $$ 1 $$ unit down. It is always increasing and curving downward (concave downward).
The function $$ T(x) = \ln(x+8) - 1 $$ is strictly increasing with no local maxima or minima. It is concave downward everywhere in its domain $$ x > -8 $$.

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